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Theoretical and Physical 









J 7 w 


THE following work on Chemical Statics reproduces the 
lectures given by me in the winter and summer semesters, 
1897-98, at the University of Berlin, under the title 
1 Selected Chapters in Physical Chemistry/ It is a second 
part to the ' Chemical Dynamics,' but at the same time is 
as far independent as possible, and deals with the methods 
which yield conclusions as to molecular magnitude, struc- 
ture, and grouping, and consequently bring into life our 
present conceptions on the structure of matter. The theory 
of solutions and stereochemistry are treated especially fully, 
in accordance with the present interest attaching to them. 
That Part I of the ' Lectures on Theoretical and Physical 
Chemistry ' has already been translated into French and 
English may perhaps be regarded as a recommendation 
of the present Second Part. 


March, 1899. 








The atomic hypothesis i r 

The molecular hypothesis 13 


Molecular weight determinations by chemical methods . . 15 

Physical methods for determination of molecular weight . 16 


GASES - 16 

A. Avogadro's law 16 

B. Methods for molecular weight determination in gases 19 

1. Ordinary process ......... 19 

2. Determination of vapour density by means of the second law 

of thermo-dynamics . . . ... . .20 

C. Results 22 

1. Connexion with the atomistic hypothesis .... 22 

2. Confirmation and testing of atomic weights . . .23 

3. Nature of the elementary molecules, and polymerism . . 24 
Polyatomicity. Hydrogen ... . . .24 

Monatomic elements. Mercury ...... 25 

Allotropy of the elements. Ozone and oxygen ... 26 
Dissociation of the elements at high temperature. The 

halogens .......... 26 

Polymerism 27 





A. The theory of dilute solutions 28 

1. Henry's law and the constitution of the dissolved gas . . 28 

2. Avogadro's law for dissolved substances . . . 31 

B. Methods for molecular weight determination of dis- 

solved bodies 36 

1. Direct methods for molecular weight determination of 

dissolved bodies ......... 36 

(a) Comparison of the osmotic pressure of different solutions 

(Isotony) 36 

(6) Absolute measurement of osmotic pressure ... 39 

2. Indirect methods for molecular weight determination . 41 

(a) The cyclic process can be carried out at constant 

temperature 42 

Molecular weight determination by vapour pressure 

measurement ........ 42 

Deduction of the law of diminution of vapour pressure 

without thermo-dynamics . . . . .43 
Thermo-dynamic deduction of the law of diminution of 

vapour pressure 46 

Abnormal values for abnormal vapour densities . . 47 

Accuracy attainable in measuring vapour pressures . 49 

Molecular weight determination by lowering of solubility 50 
Ratio of partition . . . . . . .53 

(b) The cyclic process cannot be carried out at constant 

temperature ........ 54 

Molecular weight determination by lowering of the 

freezing point ........ 54 

Molecular weight determination by the rise of boiling 

point 57 

Molecular weight determination by means of the change 

of solubility with temperature .... 58 

C. Results 59 

1. Simple molecular magnitude of dissolved bodies ... 59 
Larger molecules. Agreement with the results of gas density 

measurements ...... 59 

Larger molecules in hydroxylic compounds .... 60 

2. Development of the stereochemical conceptions . . .61 

3. Abnormal results for isomorphous compounds . . .61 

4. Abnormal results for electrolytes 62 

The theory of electrolytic dissociation ..... 62 

Electrolytes which follow Ostwald's law of dilution . . 66 

Electrolytes which do not follow Ostwald's law of dilution . 67 




A. Qualitative considerations 70 

B. Quantitative results 72 

1. Isomorphous mixtures of electrolytes ..... 73 

2. Crystalline mixtures of non-electrolytes .... 74 

Crystalline solid solutions . ..... 75 

Isomorphous mixtures ....... 76 

3. Amorphous solid solutions ....... 77 

77. MOLECULAR STRUCTURE (Isomerism, Tautomerism). 

A. Determination of constitution on the basis of the 

valency of the elements combined .... 83 

B. Determination of constitution from formation out of, 

and conversion into compounds of known 

structure 85 

Intramolecular atomic displacements ..... 88 


A. Determination of configuration by the number of 

isomeric derivatives 91 

1. Constitution of benzene 91 

(a) Benzene gives only a single monosubstituted derivative 92 

(5) Existence of only three bisubstituted derivatives . . 94 
(c) Benzene substitution products not optically active . 94 

2. Determination of position in the benzene derivatives . . 97 

B. Determination of relative distances in the molecule . 98 

1. Mutual action of different groups ...... 98 

2. Mutual influence of different groups 100 

C. Stereochemistry 102 

r. The asymmetric carbon atom and separation into optical 

antipodes . . . . . . . . . .104 

(a) Methods based on the phenomena of solution . . 107 
(a) Spontaneous separation ...... 107 

(j8) Separation by forming salts with active acids and 

bases . . . . . . . . .114 

(6) Methods of separation depending on chemical action. 

Separation by means of enzymes and organisms . 115 

2. Simple carbon linkages and compounds with more than one 

asymmetric carbon atom . . . . . . .116 

The principle of free rotation . . . . . .116 

Number of isomers in multiple asymmetry . . .117 

Inactive undecomposable type . . . . .118 

Several asymmetric carbon atoms . . . . .119 



3. Double linkage and ring formation . . . . .121 

Carbon double linkage 121 

Ring formation . . . . . . . .124 

4. Stereochemistry of other elements . . . . .124 

Concluding remark on stereochemistry .... 125 





A. The stable modification must have the smaller vapour 

pressure and the smaller solubility . . . .134 

B. The stable modification must have the higher melting 

point 134 

C. Possibility of a transition temperature . . . .135 

D. When there is a transition temperature, the modifica- 

tion stable at low temperatures is formed from the 
other -with evolution of heat 136 

E. Polymorphic modifications have a constant ratio of 

solubilities, proportional to the differential coeffi- 
cients of the saturation pressures, in solvents which 
take up so little of the substance, that the laws of 
dilute solutions are applicable 137 


A. Relative position of molecule centres in the crystalline 

figure 139 

1. The fundamental law of geometrical crystallography (F. C. 

Neumann) .......... 139 

2. Attempt at explanation of the geometrical law by arrange- 

ment of molecule centres (Frankenheim, Bravais) . -140 

3. Relations of symmetry in crystals . ..... 141 

4. Sohncke's point systems . . . . . . . 145 

5. The fundamental law of physical crystallography . . .146 

6. Molecular compounds . . . . . . . -147 

B. Orientation of the molecules in the crystal . . .148 

1. Relations between symmetry in the crystal and in the 

molecule 148 

2. Influence of changes in the molecule on the crystalline form 151 


IN the inevitably arbitrary division of any subject it is 
well to choose so that it may easily be seen where each part 
belongs. For this reason the treatment adopted by Lothar 
Meyer in the later editions of his Modern Theories of Chemistry 
seemed to me appropriate for my lectures: in it the whole 
is divided into Statics and Dynamics. Statics then deals with 
single substances, i.e. with views on the structure of matter, 
the conception of atoms and molecules, and on constitution so 
far as the determining of molecular configuration. Dynamics 
is devoted to the mutual actions of several substances, i. e. to 
chemical change, affinity, velocity of reaction, and chemical 

To these I have added a third section, in which the chief 
object is the comparison of one substance with another, and 
consequently the relations between properties both chemical 
and physical and composition. 

In preparing for the press I have preserved this arrange- 
ment, only making a change in the order in accordance with 
the development of chemical science in the last ten years. 
Until then Dynamics, that is the study of reactions and of 
equilibrium, took a secondary place. But lately, and espe- 
cially since the study of chemical equilibrium has been related 
to thermo-dynamics, and so has steadily gained a broader and 
safer foundation, it has come into the foreground of the 
chemical system, and seems more and more to belong there. 


The following arrangement is therefore chosen as an experi- 
ment : 

First Part : Chemical Dynamics. 
Second Part : Chemical Statics. 

Third Part : Kelations between Properties and Con- 

The logical advantage gained in this way is essentially that 
in the First Part it is possible to proceed without any hypo- 
thesis on the nature of matter, only the molecular conception 
being made use of. Not till the Second Part does the atomic 
hypothesis come to the front, and with it problems of con- 
figuration. Finally comes the still very obscure problem of 
the relation of one body to another. 

There are two points, however, that should be referred to. 
From the logical side it may be objected that Statics is con- 
cerned with the simpler problem, since it deals with single 
substances at rest, whereas Dynamics deals with a complex of 
substances in action. This objection, however, has less force 
when one remembers that the single substance corresponds to 
the state of equilibrium following a completed reaction and 
indeed the simplest form of equilibrium and accordingly 
Part II is devoted to the more detailed study of this final 

From the paedagogic point of view, placing Dynamics first 
can be dubious only to those chemists who are not well pre- 
pared in Physics, and consequently have not mastery over the 
chief lines of their own subject. 

The treatment chosen corresponds with that I have followed 
in teaching. It consists essentially in developing each generali- 
zation from a specially chosen concrete experimental case. On 
this follow an exhibition as far as possible graphic of the 
leading results, the conclusions drawn, and, lastly, theoretical 
remarks on the generality and applicability of the conclusions. 


Contents and arrangement. As explained in the intro- 
duction, chemical statics treats a single substance as a case 
of equilibrium of the simplest kind, occurring on completion 
of a reaction. It is, in the first instance, defined by its 
qualitative and quantitative composition. But as, for the 
same composition, it is possible to have different states 
of equilibrium, e.g. polymerism and isomerism, a closer 
characterization is necessary. The external features, apart 
from hypothesis, whether physical, such as heat of forma- 
tion, density, optical rotation, crystalline form, and so on, 
or chemical, as affinity, velocity of reaction, and so on, have 
not up to the present afforded any means of predicting 
the conditions of such polymerism or isomerism. The case 
is different if molecular and atomistic theory be taken as 
the starting-point. We must therefore treat chemical 
statics in this sense too, i. e. we must develop the views 
as to the internal structure of matter, after saying a word 
on the basis of the molecular and atomistic treatment. 

The atomic hypothesis, it is well known, is an attempt 
at explaining the peculiar relationships observed in the 
quantitative composition of compounds. Three funda- 
mental laws have proved to be strictly true : 

i. The law of constant composition, which amounts to 
saying that a compound, however prepared, has always 


the same composition, e. g. silver chloride always contains 
75.26 VAg. 

2. The law of multiple proportions, according to which 
two elements that come together in several compounds 
occur in the same ratio, or in ratios that are related by 
means of w T hole numbers ; e. g. compare 

Silver chloride (with 24-74 % Cl and 75-26 / o Ag) and 
Silver chlorate (with 18-54 % 01 and 56.40 % Ag) ; 
the ratio of chlorine to silver is the same in the two, 

24-74 : 75-26 = 0-329 and 18-54:56-40 = 0-329. 
On the other hand, compare 

Chlorine iodide (21-84 % Cl and 78-16 % iodine) and 
Chlorine tri-iodide (8-52 % Cl and 91-48 / o iodine); 
the two ratios of chlorine to iodine are in a whole number 
proportion, for 

21-84 : 78-16 = 0-279 and 8-52 : 91-48 = 0-093 = ^x0-279. 

3. The law of equivalents, which expresses that two 
elements combine with equal quantities of a third, in 
a ratio equal to that in which they combine with one 
another, or in a ratio related to the latter by a whole 
number. If, e. g., we compare 

Silver chloride (24-74% Cl and 75-26 % Ag), 
Silver iodide (54-04 % I and 45-96 % Ag), with 
Chlorine iodide (21-84% Cl and 78-16 % I), 
the ratio of the amounts of the elements combined with 
equal amounts say 75-26 parts of silver is : 

24-74 : 54-04 x |^ = 0-279, 

i.e. the same as in the direct combination of chlorine 
and iodine. 

If we had chosen chlorine tri-iodide (with ratio 0-093) 
as instance, then obviously the integral ratio 3 : 1 would 
have been found. 


These three fundamental laws are explained on the 
atomistic hypothesis. Assuming that the elements chlorine, 
silver, and iodine occur in discrete particles of fixed mass 
atoms (chlorine Cl 35-5, silver = Ag = 1 08, iodine = I 
= 127) we have : 

1. In the symbol AgCl for the particles of which silver 
chloride consists, the expression for its constant composition. 

2. In the symbol AgC10 3 for silver chlorate the expression 
that here chlorine and silver combine in the same pro- 
portion as in silver chloride ; whilst the formulae IC1 and 
ICL or generally I. m Cl w and I^Cl g refer directly to the in- 
tegral relation between the proportions in which iodine 
and chlorine occur in the later example: 

ml >I m p 

- : = ~ :- = ma : np. 

~ - 
qCl n q 

3. The constant or multiple ratio of combination between 
chlorine and iodine indirectly (combined with equal 
amounts of silver) and directly is equally expressed by 
the three general symbols 

AgpCl,, A gr l s) I W C1 M , 

since the ratios of the halogens combined with the same 
amount of silver and the direct ratio of combination are 
given by 

qCl si qrCl . nCl 

^-r : r = - r and -=- , 

r - r -=- 
rAg psL ml 


grCl nCl 

- T- : =- = arm : pen. 
psL ml 

It may be added that the assumption of the invariability 
of these atoms in quantity and quality is a simple expres- 
sion of the empirical law of the conservation of mass and 
of the invariability of the elements. 

The molecular hypome&is is a direct consequence of the 
assumption of atoms, and so rests also on a chemical basis. 


It is very important, however, that it was developed inde- 
pendently in the region of physics, and that the relations 
between molecular weights so formed could be brought into 
perfect harmony with those obtained on chemical grounds. 
Starting from these hypotheses on the nature of matter, 
and bearing in view the aim of explaining differences of 
properties when the qualitative and quantitative compo- 
sition is the same, we have first to consider molecular 
weight, and the special kind of isomerism that is referred 
to differences of molecular weight, i. e. polymerism. Next 
comes the internal structure of the molecule, with the 
differences so arising, and associated with the phenomenon 
of isomerism in the narrower sense. Finally comes the 
arrangement of molecules to form crystalline figures whose 
differences constitute the phenomenon of polymorphism or 
physical isomerism. Accordingly the following division 
is adopted: 

I. Molecular weight and polymerism. 
II. Molecular structure and isomerism. 
III. Molecular grouping and polymorphism. 

tfl^ERS 1 


As mentioned above, the molecular hypothesis rests upon 
a basis both chemical and physical. Accordingly the 
methods for the determination of molecular weight belong 
partly to chemistry, partly to physics. 

Molecular weight determinations by chemical methods 
start from the atomistic hypothesis, according to which the 
molecule is built up of atoms ; e. g. if a compound contain 
p, q, and r per cent, of carbon (C = i2), hydrogen (H=i), 
and oxygen (O=i6) respectively, the ratio of the number 
of atoms a, 6, and c is given by 

-, p r 

a : o : c = : q : p 9 
12 16 

in which a, b, and c must be whole numbers. This does 
not, however, allow of fixing a, b, and c, for if the least 
integral relation has been calculated all the formulae 

(C H, cg n , 

in which n is unity, or some other integer, would satisfy 
the conditions. 

To determine n therefore requires a second criterion, 
which in the region of chemistry mostly leads with suf- 
ficient probability to a decision, but so far not with absolute 
certainty. It is supplied by the chemical behaviour com- 
bination and reaction which renders certain molecular 
weights probable. Thus taking acetic acid 

(CH 2 0) B) 

and remembering that it arises by oxidation of alcohol, for 
which the simplest possible formula is C 2 H 6 0, and forms 


a silver salt C 2 H 3 Ag0 2 containing at least two carbon 
atoms in the molecule, n = i for acetic acid is practically 
excluded, and n=z rendered very probable. 

These chemical methods, though mostly sufficient in 
practice, are exposed to the logical objection that they 
depend on products of reaction, and that during the reaction 
a total transformation of the molecular complex might 
occur ; e. g. the formation of acetic acid might be 

C 2 H 6 + 20 = 2CH 2 + H 2 0, 

and on the other hand can at most only give a probable 
minimum size for the molecule; it is therefore fortunate 
that the physical methods are suitable for directly testing 
the molecular weight of the bodies in question. 

In the following collection of physical methods for 
determination of molecular weight, a selection has been 
made in accordance with the plan of the work. On the 
one hand are the methods based on Avogadro's law, and 
applicable to gases and solutions ; on the other are methods 
applicable to liquids, of which that of Eotvos, Ramsay, and 
Shields, based on the change of surface tension with tem- 
perature, is an example ; on account of the empirical 
character of the latter they will be found in Part III, 
which is concerned with the relations between properties 
(e. g. capillary constant) and composition (and therefore 
molecular weight). We shall therefore confine ourselves 
here to : 

i. Molecular weight determination in dilute gases. 

2. solutions. 

3. solid solutions. 



A. Avogadro's Law. 

It is well known that the law on which the determination 
of molecular weights in gases is based, as stated by Avogadro, 
is to the effect that equal volumes of dilute gases, measured 


under the same temperature and pressure, contain equal 
numbers of molecules, and therefore the molecular weights 
of the gases are in proportion to the weights of those 

The law in question, which, when first stated, was a happy 
combination of known facts, received confirmation in that 
it allowed of predicting new ones, such as the dissociation 
of ammonium chloride on volatilization. But it may also 
be deduced from the kinetic theory. As we shall bring 
forward the confirmations of the law later, under the head 
of 'results/ we will here begin with its theoretical 

The kinetic theory of gases, treating gases as consisting 
of small, perfectly elastic spheres, involves Avogadro's law 
as the limiting case of extreme dilution. When the 
dilution is sufficiently great, the attractions between the 
molecules cease to be of account, and the volume of 
the molecules may be neglected in comparison with the 
total volume occupied by the gas. The deduction of 
Avogadro's law from the kinetic theory in the general 
case is complicated, and for the purpose of this work it 
is sufficient to follow it out under special simplifying 

In the first place, for simplicity it may be assumed that 
the velocity of all the molecules is the same, viz. metres 
per second. Secondly, that the 
movement inside the cubical 
vessel of V cub. metres ca- 
pacity (Fig. i) that we shall 
consider, is so distributed that 
the molecules move only at 
right angles to the bounding 
planes, and equally so, there 
being always one-sixth of the rio . 

N molecules moving forwards, 

backwards, right, left, up, or down. Hence there strike 
unit surface (one sq. metre) in unit time (one second), one- 



sixth of the molecules contained in a parallelepiped at 
right angles to the surface, and C metres long ; i. e. 



They are repelled with equal velocity; so that the whole 
have in the second suffered a change of velocity of 2(7 
metres per second. The force needed for that (P kilo- 
grams pressure per sq. metre) may, according to the law 

force = mass x acceleration, 
be calculated as 

P- , j- -~ .-u -r\TT 

= -^MxzC = T7 - or PV= 
OK 3 V 

where M is the mass of a molecule. 

For two gases at the same temperature and pressure we 
have consequently : 

The third condition, equality of temperature, includes, 
however, a further relation, for it may be shown that 
equality of temperature corresponds to equality of mean 
kinetic energy of the molecules, or 

(2) \MW = \M*, 

from which by combination with (i) we get 

^ = N 2> 

that is an equal number of molecules for equal volume, 
pressure, and temperature. 

Before we describe the methods in which Avogadro's law 
may be applied, we may conveniently put it in a single 
mathematical form which expresses also the laws of Boyle 
and Gay-Lussac. It is well known that the last two may 
be represented by 

If the molecular quantity of different gases be taken, then 
according to Avogadro's law, as just stated, for equal 
pressure (P) and temperature (T), the volumes (V) will 


also be equal, which amounts to saying that R referred to 
the molecular quantity has the same value for all gases. 

Taking 2 kilograms of hydrogen as basis for calculating 
this value of R, we have for atmospheric pressure and 


P = 10333 kg/sq. m. T = 273 V = cub. m. 

since one litre of hydrogen at o and atmospheric pressure 
weighs 0*08956 gm. Therefore 

R = 845-2. 
Since this number happens to be approximately twice 

the value of the calorie in kilogrammetres (-T- = 423) the 

expression for the laws of Boyle, Gay-Lussac, and Avogadro 
simplifies, approximately, to 

B. Methods for Molecular Weight Determination in 


i. Ordinary Process. 

The application of Avogadro's law, formerly the only 
means of determining molecular weights, has, since the 
introduction of the simple methods based on the theory 
of solutions, been practically confined to Victor Meyer's 
process 1 . 

, As in all such methods, the procedure consists essentially 
in measuring the volume (V) of a known weight (G) as gas 
or vapour under a definite pressure (P) and temperature 
(T). An equal volume of hydrogen would, under these 
circumstances, weigh 

273 PV 

0-08956 PVx-tp = 24-45^- 
if weight, volume, pressure, and temperature are given in 

1 Berl. Ber. 30. 1926. 
B 2 


grams, litres, atmospheres, and in absolute centigrade 
measure respectively. Since under the specified circum- 
stances there is, according to Avogadro, proportionality 

between total weight and molecular 

weight, we get : 


where M is the desired molecular weight, 
and 2 that of hydrogen (see p. 24). 

In Victor Meyer's method a cylindrical 
vessel b (Fig. 2) of about 200 c.c. capacity, 
ending in a long neck with a side tube af, 
is filled with air or nitrogen, and brought 
to a convenient constant temperature. If 
now the weighed quantity of substance 
be introduced into the cylinder by means 
of the dropping arrangement d, the vapour- 
produced expels an equal volume of air 
which passes through the side tube of 
and is collected in the graduated tube g 
over water. 

The great advantage of this process is 
that it allows of the evaporation being 
performed at a very high temperature, 
which heed not be exactly determined. 
Ignorance of this temperature has no in- 
fluence on the result, as it is the tempera- 
ture of the graduated tube, not that of 
the cylinder, that is needed to calculate 
Fig. a. the molecular weight. 

2. Determination of Vapour Density by means of the 
Second Laiv of Thermo-dynamics. 

We will mention here a second method which, though, 
so far, of no consequence in the practical measurement 
of molecular weights of gases and vapours, may be of 


interest at the present time in its special application to 

The method is based on the application of the second 
law, which, as applied to the process of evaporation, was 
given (Part I, p. 19) as 

where A is the thermal equivalent of work = 1 ^ B , V the 
increase in volume on evaporation, in cub. metres, of 
a quantity (say i kilogram) that absorbs q calories, P the 
pressure of the saturated vapour in kilograms per sq. metre. 
A vapour density and therefore a molecular weight 
determination may be obtained from this equation, since 
the volume of the vapour itself may be taken as V when 
the dilution of the saturated vapour is sufficiently great, 
so that the measurement of volume is based upon that of 
the change of vapour pressure with temperature, and the 
latent heat of evaporation. Taking water as an example, 
with the data : 

P 10 = 9-14 mm., P 20 = 17-363, <?io = 5 8 4cal., 
we get, without integration, using the formula 

_ 584x10 

ATAP" 388 x 8-233 x 13-6x4 

where 13-6 is the density of mercury. Now 77 cub. metres 
of hydrogen at 13-2515 mm. and 15 weigh 

77 x 0-08956 x ^p- 5 x 2 || = 0-114 kg. 

700 2oo 

So that the molecular weight of water vapour according to 
Avogadro is 

M : 2 = i : 0-114. 

M= 17-6, 

which is close to the known value H 2 O = 18. 

In the same way the observations for acetie acid show 
an abnormal molecular weight. We may perform the 
calculation in a simpler way by using, instead of the 


kilogram as before, the molecular weight M as obtained 
from the vapour density. Then 

.dP MqdT .- 2T 2 dP 
AVP = *T, so that - = -- or M= 

Introducing the values 

P 10 = 6-6 mm., P 20 = n-6 mm., q = 85 cal., 

we have 

.... 2X288 2 x<5 

M = 7T = J07 

85 xg-i xio 

instead of 60 ( = C 2 H 4 2 ),in accordance with the conclusion 
from direct vapour density measurements, that saturated 
vapour of acetic acid has a nearly double molecular weight, 
and consists mainly of double molecules. 

C. Results. 
i. Connexion with the Atomistic Hypothesis. 

The fundamental scientific interest of the results 
obtained by application of Avogadro's law lies in the 
complete connexion that can be established between the 
chemically determined atomic weights and the molecular 
weights arrived at physically. 

The necessary conclusion from atomistic reasoning, that 
in reactions the molecules concerned must be in whole 
number relationship, as e. g. when methane (CH 4 ) is burnt 
in nitrous oxide (N 2 O) 

CH 4 + 4 N 2 = C0 2 + aH 2 + 4N 2 , 

agrees with Gay-Lussac's law that in reactions the volumes 
of gases concerned are in such a relationship. According 
to Avogadro the ratios in question may be read off the 
equation directly, in the form of the coefficients occurring 
in it, as each molecular symbol corresponds to the same 
volume, so that 

Vol. CH 4 : Vol. N 2 : Vol. C0 2 : Vol. H 2 O : Vol. N 

= 1:4:1:2:4. 


This relation occurs most simply in the case of . the 
elements. The ratio of their molecular weights, i. e. of 
their gas densities, must either be the same as that of their 
atomic weights, or, if the molecules are built up out of 
different numbers of atoms, be in integral relation to those- 
Exact proof of this is rendered difficult by the fact that 
Avogadro's law is only exactly true for infinite dilution. 
A recent extrapolation l in that direction leads to the 
conclusion that the densities of hydrogen, nitrogen, and 
oxygen on infinite dilution are in the ratio 

1-0074 : 14-007 : 16, 

whilst Ostwald gives for the atomic weights : 
10032 : 14-041 : i6. 

2. Confirmation and Testing of Atomic Weights. 

The determination of atomic weights by purely chemical 
means is attended with an uncertainty, which may be 
noticed in the history of the atomic weights, and may 
be illustrated in the following way. Choosing as unity 
H = i, or rather O = 16 as better suited for analytical 
purposes, the analysis of e.g. beryllium oxide with 36-3 
per cent, beryllium is not conclusive as to the atomic 
weight of beryllium, but leaves it dependent on the formula 
of the oxide. If it is regarded as BeO we get 
Be : O = 36-3 : 63-7 = x : 16 ; 
x = 9-1 ; 

whilst if Be 2 3 is chosen and both formulae have had their 
supporters it follows that 

2 Be : 36 = 36-3 : 63-7 = zx : 3 x 16; 
x= f X9-i. 

The decision here and in corresponding cases has in the end 
rested on determinations of molecular weight. The deter- 
mination is not possible for the oxide on account of its 

1 D. Berthelot, Compt. Rend. 126. 954. 


non-volatility, but can be made with the chloride. The 
molecular weight of the latter, according to the respective 
assumptions, would be 

-Wi'eci, = 9-1 + 71 = 80-1, Jf Be ci 3 = I x 9-1 -4- 106-5 = 120. 

The former number was found by Nilson and Petterson l 
and the atomic weight 9-1 so finally settled. The only 
change possible would be a halving, &c., of this atomic 
weight, so that the method gives a maximum value, below 
which there is neither chemical nor physical reason for 
going, and which has, moreover, received confirmation in 
another way (p. 26). 

3. Nature of the Elementary Molecules, and Polymer ism. 

Poly atomicity. Hydrogen. One of the leading results 
of molecular study of the elements is the fact that their 
molecules usually consist of more than one atom. Thus, 
the two facts that prove this for hydrogen are : 

The density of hydrochloric acid is 18-25 referred to 
hydrogen, and it contains 2-74 per cent, of hydrogen. 

Since the densities, according to Avogadro's law, are in 
the ratio of the molecular weights, we conclude from these 
data that the ratio between the amounts of hydrogen 
present in the hydrogen and hydrochloric acid molecules is 

i : 18-25 x-^ = a : i. 

Thus there is twice as much hydrogen in a hydrogen 
molecule as in one of hydrochloric acid, and as the latter 
cannot contain less than one atom of hydrogen, there must 
be at least two in the hydrogen molecule. Such reasoning- 
has in no case, so far, led to the necessity of assuming more 
than two atoms in the hydrogen molecule, so it may be 
regarded as diatomic, with the symbol H 2 . For that reason 
hydrogen, with M= 2, is usually chosen as unit for molecular 
weight determinations. 

1 See also Rosenheim and Woge, Zeitschr.f. anory. Chem. 15. 283. 


Most of the elements studied are, like hydrogen, diatomic ; 
only a few have more atoms, as phosphorus P 4 , arsenic As 4 , 
sulphur S 8 . 

Monatomic elements. Mercury. Especially interesting 
are the cases in which the elementary molecule is, according 
to the above reasoning, to be regarded as monatomic. This 
was first shown to be probable for mercury ; since then 
other metals (zinc, cadmium, potassium, sodium) have been 
studied in the state of vapour, and appear to behave 
similarly; and the newly discovered elements, argon and 
helium, belong to the same category. Mercury, moreover, 
was the subject of an important experiment by Kundt and 
Warburg, which, by the aid of a deduction from the kinetic 
theory, confirms the conclusion of monatomicity. The 
experiment refers to the ratio between the specific heats 
at constant pressure and constant volume. The latter has 
obviously the smaller value, and in the case of monatomic 
gases represents the increase in kinetic energy of the 
molecular movement ; in polyatomic gases it includes also 
an increase in the atomic movements, more difficult to 
calculate. According to the previous equation 

PV= NMC 2 
the kinetic energy 

INMC* = f PF, 

so that its increase per degree is 


where a = - - 


If free expansion (at constant pressure) takes place, we 
are concerned with the specific heat at constant pressure, 
and additional heat is required to perform external work, 
which obviously amounts, per degree, to 


The ratio of the two specific heats is therefore 

faPF+aPF: f aPF =5:3 = 1-67, 


whilst for polyatomic gases it is less, since the term fa 
will be greater on account of the kinetic energy of the 
atomic movements. It is well known that measurements 
of the velocity of sound gave the expected value 1-67 for 
mercury vapour, and lower values for polyatomic gases. 

As according to p. 24 the atomic weights so far found 
are strictly to be regarded as maxima, we have now an 
indication from thermal reasoning that, at least for 
mercury, the maximum has not been placed too high. 

Allotropy of the elements. Ozone and oxygen. A third 
important result of molecular weight measurements of the 
elements has been to show, on grounds of the molecular- 
atomistic hypothesis, the existence of the same element in 
different, so-called allotropic, forms. Oxygen and ozone 
form the most striking example. Soret proved that ozone, 
on conversion into oxygen, increased in volume by one-half, 
so that oxygen being diatomic, ozone is to be regarded as 
O 3 , the equation 

20 3 = 3 2 

indicating the increase in the number of molecules by one- 
half, and according to Avogadro, increase of volume in 
the same ratio. 

Dissociation of the elements at high temperature. The 
halogens. A very welcome confirmation of the above views 
on the polyatomicity of the elements is given in the proof 
of the possibility of decomposing those molecules. Even 
sulphur, which at low temperatures appears as S 8 , shows an 
abnormal expansion up to 1,000, leading by then to a 
density corresponding to S 2 . A further step in this 
direction was accomplished by Victor Meyer for the 
halogens, since he showed that iodine, which under 600 
has a density corresponding to the diatomic molecule I 2 , 
becomes halved in density above 1,400, with formation of 
rnonatomic molecules or free atoms. 

It may be added that in this direction lies the possibility, 
if any, of decomposing the elements themselves : the mona- 
tomic molecule constitutes the last stage towards the 


undecomposed fundamental material ; if the structure can 
be still further simplified, then a further decomposition is 
attainable. This is exactly the point of view adopted by 
Victor Meyer in his latest researches. In his Liibeck 
address on 'Problems of Atomistics ' (1896) he referred to 
this aim. After that, and until the end of his life, he 
occupied himself with experiments in this direction l , and 
had in view molecular weight determinations up to 2,000. 
The glass vessels had long been replaced by others of 
porcelain, and this material by platinum or the still more 
infusible alloy with 25 per cent, iridium. Finally vessels 
of magnesia were prepared, which, in a lime oven, stood the 
temperature of 2,000 obtained by burning graphite in 
oxygen. But no further decomposition than that into 
monatomic elements was reached in the case of any 

Polymerism. The difference in molecular weight associ- 
ated in the elements with differences of properties occurs 
also, it is well known, in compounds of the same composi- 
tions, especially in organic chemistry, and is then called 
polymerism. Whole series of similarly composed bodies, 
such as 

are known, in which n varies from 2 to 30, accompanied by 
a corresponding change in vapour density and chemical 
behaviour, as e. g. in combination with bromines, the bodies 

(CH 2 ) n Br z , 
of different composition, arise. 


Since the possibility of molecular weight determinations 
of gases and vapours was shown by Avogadro, the modern 
development of the theory of solutions rendered the same 

1 Berl. Ber. 30. 1926. 


thing possible for these latter substances. We will therefore 
next develop that theory, and then, in connexion with it, 
give the methods for determination of molecular weight, 
with their results and applications. 

A. The Theory of Dilute Solutions. 

Whilst for the determination of molecular weight in 
dilute gas Avogadro's law can be applied directly, for dilute 
solutions the most varied methods, based on measurements 
of freezing points, boiling points, and vapour pressures, are 
in use. All these methods, however, may be brought into 
connexion with, and regarded as deductions from a law 
exactly corresponding to that of Avogadro, but which refers 
to osmotic instead of gas pressure; the law states, in fact, that 
solutions that exert the same osmotic pressure at the same 
temperature contain equal numbers of dissolved molecules in 
unit volume. This law may be arrived at in various ways. 
We shall here start from the molecular properties of known 
gases, and follow the deduction through gaseous solutions. 

i.. Henry's Law and the Constitution of the Dissolved Gas. 

Take any gas of known molecular constitution, e. g. 

nitrogen (N. ? ) and a liquid, water, in which it is capable of 
solution, and consider the question whether 
the dissolved nitrogen corresponds to the 
formula N 2 or possibly is present as N or 
N 4 or as a hydrate. We have then to con- 
sider the equilibrium that is set up when the 
water B (Fig. 3) is saturated with the nitrogen 
Fig. 3. A under the existing conditions of tempera- 

ture and pressure. Kinetically considered, 

this equilibrium is based on the fact that, in unit time, as 

many gaseous molecules enter the solution B as leave the 

latter l . 

1 In the following considerations the vapour of the solvent, present 
in A, may be supposed absent, by covering B with a membrane permeable 


But from this a conclusion may be drawn if the nitrogen 
present in B consist entirely of molecules N 2 , i. e. is of the 
same constitution as in the gas. For if the quantity per 
unit volume in A is doubled, that in B must be so too, for, 
with sufficient dilution, this change would double both the 
number of gaseous molecules passing from A into B and 
the number of dissolved molecules passing from B into A. 
So that in this case the concentration C in the solution is 
proportional to the pressure P of the gas 

a law that, under the name of Henry's law, is known to 
be applicable to the majority of gaseous solutions. 

Let us now suppose a different case : first, that nitrogen 
is present in water as N, not as N 2 ; then the law of absorp- 
tion becomes different also. In order to deduce this modified 
law, let us assume that in the gas too a certain quantity, 
even though vanishingly small, of nitrogen occurs as N ; 
then between that and the nitrogen in the water there 
must be an equilibrium as in the preceding case, and pro- 
portionality as before 


where P N is the very small partial pressure of the gaseous 
nitrogen in the form N. This partial pressure is not, 
however, proportional to the total pressure P, since the 
equilibrium in the gas between the two forms of nitrogen, 
according to the equation 

is subject, according to the laws of chemical equilibrium 
(cf. Part I, p. 109), to the relation 

where ./V a and PN are the partial pressures of the nitrogen 
in the forms N 2 and N respectively, and consequently P y 

to nitrogen, but not to water vapour; although if the vapour were 
present the reasoning would still hold. 




differs only by a vanishingly small amount from the total 
pressure P. Hence 

P = KP\ and C = ^P N = AVP, 

where k is a constant, equal to 

In this case, then, the concentration of the dissolved 
substance would not be proportional to the pressure of the 
gas, but to its square root, which would imply a very wide 
departure from Henry's law: quadruple pressure would 
not involve a quadrupling of the concentration, but only 
a doubling. 

A different molecular constitution, such as N 3 or N 4 , would 
introduce a corresponding change in the law, so that there 
only remains to consider the case of hydrate formation, 
such as N 2 . H 2 O. This case may be treated in a similar 
manner, by assuming the presence in the gas of an 
extremely small quantity of the hydrate, or in the solution 
of, besides hydrate, an extremely small quantity of nitrogen 
not combined with water. Both assumptions lead to the 
same result; by assuming a small amount of hydrate in 
the gas, we bring into play the equilibrium 

N 2 .H 2 0^N 2 + H 2 0, 
with the condition 

PN 2 H.H 2 O = -2* K s > 

in which PN 2 .H 2 o and P Nj8 are the partial pressures of the 
hydrate and of free nitrogen respectively. Since obviously 

PN Z = -P -PNaHaO 
the equation may be replaced by 

-PN 2 .H 2 = &!*> 


Since according to what preceded, the partial pressure of 
the hydrate is proportional to the concentration of the 


dissolved nitrogen, we get a ratio between pressure and 


C = kP, 

exactly as if the nitrogen were dissolved in the form N 2 
instead of N 2 . H 2 O. Hence the presence of a hydrate has 
no influence on the law of absorption ; and generally that 
of any hydrate N a (H 2 O) 6 does not differ from solution of 
the water-free molecule N a . 

We may therefore conclude that gases which on solution 
follow Henry's law, possess the same molecular weight in 
the solution as in the gaseous state, with only the possible 
/ exception that hydrate formation, always without change 
in size of the gaseous molecule, may occur. As already 
remarked, nearly all the gases whose absorption has 
been studied belong to this category: N 2 , H 2 , O 2 , CO 2 , 
CO, N 2 O, CH 4 , C 2 H 4 , H 2 S, NO, C 4 H 10 , C 2 H 6 in water and 
alcohol ; C0 2 in carbon disulphide and chloroform ; C 2 H 2 
in acetone 1 . Indications of a different behaviour occur 
with NH 3 and S0 2 in water, while HC1 in water departs 
so far from Henry's law that an entirely altered molecular 
magnitude on solution is to be concluded. 

2. Avogadros Law for Dissolved Substances. 

What for our purpose is essential in the preceding 
argument is not that the molecular weight of some ten or 
twelve gases in solution is known, but the principle that 
if Henry's law of absorption answers to the facts, then 
a gas does not change its molecular magnitude on going 
into solution. On account of the limited data available on 
absorption, and especially on account of the fact that most 
substances are too involatile to give gases or vapours of 
measurable density, the principle discussed is of minor 
importance in its direct application. We have rather to 
consider the question without assuming any experimental 
data on absorption what property must any dissolved 

1 Compt. Rend. 124. 988. 


body possess as gas or vapour, in order that it may, on 
absorption, follow Henry's law ? and the answer is that the 
gas or vapour must, at equal temperature and concentration, 
exercise a pressure equal to the osmotic pressure of the 
same substance in the dissolved state. 

The proof of this may be given in a form due partly to 
me l , partly to Lord Rayleigh 2 , and partly given in a private 
correspondence with Dr. Donnan. 

Before entering on the proof, which deals with osmotic 
pressure and semipermeable membranes, i. e. membranes 
that only allow the solvent to pass, it may be remarked 
that any notion one may form as to the mechanism 
producing osmotic pressure, or the action of semipermeable 
membranes, is without influence on the reasoning. Thus 
the question whether the pressure is produced by the 
solvent or by the dissolved body can 
be left out of consideration ; so too, 
whether it is dependent on collisions 
or by attractive forces. The action 
of the membrane too, whether it is as 
a sieve, or by means of absorption, 
is indifferent. All this is the case 
because the proof to be given is based 
Fig. 4. 1 on thermo-dynamics, and is conse- 

jquently free from assumptions on the 
mechanism. Moreover it is plain that two semipermeable 
membranes cannot give different osmotic pressures, for that 
would allow of a perpetuum mobile; thus let the ring 
(Fig. 4) be supposed filled with solution on the right, and 
solvent on the left, separated by two semipermeable mem- 
branes above and below, which give osmotic pressures 
amounting to p { and p. 2 respectively. Such a difference of 
pressure p l p 2 would then give rise to a flow, which, as 
all the conditions remain unchanged, would never cease. 

After these remarks, let us proceed to the actual proof. 
By means of a reversible cyclic process carried out at 

1 Zeitschr.f. Phys. Chcm. I. 488. 2 Nature, 55. 253. 



a constant temperature T, a kilogram of dissolved 
substance X is to be removed from, say, an aqueous 
solution in the form of gas or vapour, and restored to it 

The removal of the dissolved body X is to take place 
by means of semipermeable membranes : the solution is 
separated from the un dissolved gaseous X by the partition 
be (Fig. 5), which only allows the gas to pass, whilst the 
pure solvent may be supplied through the walls a b and cd, 
on the outside of which is solvent. 
Gas or vapour pressure and os- 
motic pressure are to be kept in 
equilibrium by means of two 
pistons, above and below. Now 
i kilogram of the dissolved body X 
occupies under temperature T and 
pressure P (kg. / sq. m.) a volume 
of V cub. metres. The ratio of 
absorption is such that this vapour 
is in equilibrium with a solution 

that contains i kilogram of X in v cub. metres, which solution 
exercises an osmotic pressure p (kg. / sq. m.). If now 
both pistons be moved upwards, i kilogram of X may 
be removed reversibly from the solution, an amount of 
work being done by it 

PV=RT ....... (i) 

whilst an amount of work is performed against the osmotic 
pressure, which w^e will express with the negative sign as 

Fig. 5. 

A second process now, will restore the vapour of X to' the 
solution. First let the vapour expand to a very great 
volume V '^ and in so doing perform work 

-iT (3) 

The so diluted vapour may now be brought into contact 



with a volume v of water, a process which, in the limiting 
case in which V^ is infinite, is reversible, for under those 
circumstances the water would not absorb any of the 
infinitely dilute vapour X. Now let the piston be lowered 
and so X be brought into solution, with an expenditure of 

Here, however, P 1 has not the value given by 

but a smaller value, because a part of the vapour has gone 
into solution. This part is exactly i kilogram when the 
pressure has risen to P, and consequently, if Henry's law of 
absorption is true, when the pressure is P, amounts to 
Pi/P kilograms; the undissolved part remaining is there- 
fore i P! /P, and the pressure P may be calculated from 

so that 



Consequently the work done is 

-('"Pdv- RT[ VX ^ an<*! 
Jo ^ K I- J y, 7, + y- V 

which, if V ^ is infinitely great, becomes 

Since the total work done in a reversible cyclic process 
at constant temperature must be zero 

( + (2) + (3) + (4) = RT-pv + RT log ^f -RT log ^ = o 


pv = RT. 
By comparison with 

PV = RT, 


since T is constant, we see that for equal values of V and v 
we must have p _ ^ 

This is the result originally referred to, that any dissolved 
body which obeys Henry's law, must have such properties 
that for equal temperature and concentration, the gas 
pressure and the osmotic pressure must be equal. 

But we know, by what precedes, that a vapour dis- 
solving according to Henry's law must have the same 
molecular character in the solution and the vapour, and 
consequently we may draw all the conclusions as to 
the osmotic pressure of dissolved bodies that have been 
drawn as to gas or vapour pressure, i. e. we may apply 
Avogadro's law to solutions, making use of the osmotic 
pressure instead of the gas pressure. 

From this necessary equality of gas and osmotic pressure 
for similarity of molecular constitution in solution, it 
follows that the osmotic pressure obeys the gaseous laws, 
i. e. the laws of Boyle and Gay-Lussac. A remark on the 
nature of the osmotic pressure may be made here; if it 
follows Gay-Lussac's law, and is so proportional to the 
absolute temperature, it, like gas pressure, becomes zero 
at the absolute zero of temperature, and consequently 
vanishes when the molecular movements come to rest. 
It is, therefore, natural to look for the cause of osmotic 
pressure in kinetic grounds, and not in attractions. 

A second general remark may be made in connexion 
with the gaseous laws as applied here. In this case too 
they are to be regarded as limiting approximations, only 
strictly true for infinite dilution. They refer, on account 
of the kinetic nature of the osmotic pressure, to that part 
of it which prevails on increase of dilution ; whilst, when 
the concentration is greater, the attraction of the solvent 
becomes of more consequence, and eventually must prevail, 
since it increases in proportion to the square of the con- 
centration, while the number of molecular collisions is only 
proportional to the first power. 

C 2, 


B. Methods for Molecular Weight Determination 
of Dissolved Bodies 1 . 

As in gases a measurement of molecular weight involves 
a knowledge of the weight, volume, pressure, and tempera- 
ture in the gaseous or vaporous condition, so in solutions 
the same four data suffice, the osmotic pressure replacing 
the ordinary pressure. 

From the theoretical point of view, the first thing to 
consider is the osmotic pressure itself; but in practice 
indirect methods involving quantities related to that 
pressure occur ; we shall therefore discuss in turn the 
direct and indirect methods of molecular weight deter- 
mination for dissolved bodies. 

i. Direct Methods for Molecular Weight Deter- 
mination of Dissolved Bodies. 

(a) Comparison of the osmotic pressure of different 
solutions (Isotony). Whilst for gases the determination 
of molecular weight is an easy operation, for solutions 
a direct application of the corresponding laws meets with 
the difficulty of measuring osmotic pressure a difficulty 
due to the necessary use of a semipermeable membrane, 
i. e. a wall or boundary permeable to the solvent but not 
to the dissolved body. If the absolute value of the osmotic 
pressure is to be measured, the ' membrane ' has, in addition, 
to stand a mechanical pressure which may be great, which 
makes the experiment still more difficult ; we shall, there- 
fore, first describe the methods in which this difficulty, 
due to pressure, is avoided by observing, instead of the 
actual osmotic pressure of a solution, equality between 
two solutions of equal osmotic pressure ; afterwards the 

1 Verschaffelt, Le poids mole'culaire de I'eau el de Tiode ; Acad. des Sciences de 
Belgique. Bruxelles, 1896. Biltz. Praxis der Holekulargewichtsbestimmung, 
Berlin, 1898. 


few direct measurements that have been carried out will 
be mentioned. 

First, as to the semipermeable membrane. The peculiar 
property of allowing one substance to pass, and not the 
other, seems to depend not so much on sieve-like action 
as on the ability to dissolve or absorb, or loosely combine 
with one substance. Such a selective action occurs with 
gases, e.g. palladium 1 allows the passage of hydrogen, 
which is absorbed by it. Nernst 2 has described a semi- 
permeable membrane for liquids, based upon the same 
principle, and consisting of an animal membrane, moistened 
with water, placed between (moist) ether on one side, and 
(moist) ether, in which benzene was dissolved, on the other. 
The ether, being slightly soluble in water, passes through 
the membrane towards the side containing benzene, whilst 
the benzene, insoluble in water, fails to pass through. 
Quite recently the property of caoutchouc, noticed by 
Graham, of transmitting certain gases, such as sulphur 
dioxide and carbon dioxide, has been studied with regard 
to liquids 3 : whilst e. g. methyl and ethyl alcohol cannot 
permeate the caoutchouc, ether, carbon disulphide, chloro- 
form, benzene, &c., can, so that if one of these liquids be 
placed on one side of the membrane, and methyl alcohol 
on the other, ether, &c., passes through. Moreover, it has 
been shown that the rate at which the transmission takes 
place is in agreement with the rate at which the liquid 
in question is absorbed by caoutchouc. Finally, Tammann 
discovered in the zeoliths a material permeable only for 
water not for substances dissolved in it; these zeoliths 
are known to be hydrated silicates, which, according to 
Mallard, possess the peculiar property of absorbing and 
giving up water, without losing their crystalline form. 

Whilst those membranes are of importance for the expla- 
nation of semipermeability, an entirely different material 

1 Ramsay, Zeif.schr. /. Phys. Chem. 15. 518; Hoitsema, 1. c. 17. i. 

2 Neriist, 1. c 6. 38. 

3 Raoult, Comptes Rendus, 121. 187 ; Fusin, 1. c. 121. 794. 


has been used for actual measurements, viz, so far, mostly 
membranes occurring in animal and vegetable tissues, and 
the precipitate discovered by Traube. 

The membranes in animal and vegetable organisms, which 
offer little resistance, have proved especially useful as 
semipermeable partitions, and an example of each will be 
given here. In the first place we have the protoplasm 
sheath of the plant cell *, an elastic membrane resting 
freely against the cell-wall, being kept against it by the 
osmotic pressure of the contents. If, however, the cell, or 
a layer of cells, suitable for microscopic observation be 
placed in a salt solution of high osmotic pressure, the sheath 
contracts away from the cell-wall : the process known as 
plasmolysis occurs, as may be very well seen when the 
protoplasmic contents are coloured (Tradescantia discolor). 
Different solutions that exercise so high an osmotic pressure 
as just to produce plasmolysis, are equal in their osmotic 
action, are so-called isotonic, and therefore, according to 
Avogadro's law as extended to solutions, contain equal 
numbers of dissolved molecules in the same volume. Thus 
e.g. a solution containing 5-96 per cent, of raffinose (a sub- 
stance whose molecular weight was then unknown) was 
found isotonic with a solution of cane-sugar (C^H^O^ = 
342) containing 01 gm. mol. per litre (i.e. 3-42 / o ); hence 
the molecular weight of raffinose must be approximately 596 

3.42:5.96 = 342:;*; x = &6, 

and as for raffinose the choice lay between C 12 H 22 O n 
. 311.0 = 396 (Berthelot and Ritthausen), C 18 H 32 O 16 . 5H 2 O 
= 594 (Loiseau and Scheibler), and C 36 H 64 O 32 . ioH 2 O = n88 
(Tollens and Rischbiet), this was decisive in favour of the 
second formula, a conclusion which has since been verified 
by chemical means, since raffinose takes up water and 
decomposes into three molecules of sugars, each containing 
six carbon atoms (glucose, laevulose, and galactose) : 

1 De Vries, Zeitschr. f. Phys. Chem. 2. 440. 



A similar determination of isotony by physiological 
means, in this case, however, taken from the animal 
organism, may, according to Hamburger 1 , be carried out 
with red blood corpuscles, the solution to be studied being 
shaken up in a test tube with a couple of drops of defi- 
brinized blood. Two phenomena are then observed, 
according as the solution exercises a great or small osmotic 
pressure : in the first case the corpuscles give up their 
colouring matter to the solution ; in the second they sink 
to the bottom of the colourless liquid. Liquids which lie 
on the margin in this respect are isotonic, and so we have 
an easily available means for molecular weight determina- 

Similar isotony has been observed by Tammann 2 without 
physiological aid, by means of a precipitated membrane, 
that is a colloidal skin formed by contact of two solutions. 
The most available material yet found for this purpose is 
copper ferrocyanide, obtained by contact of potassium 
f errocyanide and copper sulphate in layers ; this was used, 
and the passage of water in one or the other direction 
observed by means of Topler's interference apparatus: 
where water enters, a rising stream in the specifically 
heavier salt solution is observed ; where the reverse 
occurs, a descending stream due to local increase in 

(6) Absolute Measurement of Osmotic Pressure. As 
already mentioned, absolute pressure measurements of 
osmotic phenomena are rendered difficult by the fact 
that it is not only necessary to prepare a semipermeable 
partition, but one that has sufficient resistance to stand 
the commonly great mechanical pressures involved. The 
organic membranes, which were found so well adapted for 
determining isotony, are for that reason useless for the 
present purpose, and consequently there are only a few 
isolated experiments that have been carried out with success, 

1 Zeitschr.f. Phys. Chem. 6. 319. 

2 Wied. Ann. 34. 299. 


| B J H 2 


I 1 

but which, however, are of the highest importance in 
testing the laws of osmotic action. 

We may first mention a measurement of Ramsay on 
gases l which is very instructive as to the mechanism by 
means of which osmotic pressure is produced. A palladium 
vessel A (Fig. 6) constituted the mem- 
brane and contained nitrogen, whose 
pressure could be measured, as shown 
in the figure. A was then surrounded by 
a stream of hydrogen at known pressure, 
and the pressure inside A rose, on account 
of the gas passing through the palladium 
partition, by an amount nearly equal to 
the external hydrogen pressure (would 
perhaps have coincided with it if the 
external atmosphere had not originally 
contained more hydrogen). Therefore the (osmotic) excess 
of pressure of the (dissolved) nitrogen present in the 
hydrogen corresponded to the pressure of the nitrogen 
alone (as Avogadro's law applied to solutions requires). 

Next we have Pfeffer's measurements of solutions. He 
used a semipermeable membrane of copper ferrocyanide, 
but gave it sufficient strength by depositing it inside a 
small porous battery-cell, the cell (moistened) being filled 
with potassium ferrocyanide, and dipped in copper sulphate. 
Diffusion from both sides then brings about the formation 
of the membrane. With such an apparatus the following 
measurements were carried out on a one per cent, sugar 
solution (i gm. in 100-6 c.c.). 

. 6. 

Temp, (t) 

T 4 


Osm. press. (P) 

o 664 atm. 
0-721 ,, 
0.716 ,, 

M = 0-813 x 


273 + t 

1 Zeitschr.f. Phys. Chem. 15. 518. 

2 Osmotische Untersuchungen. Leipzig, 1877. 


From these data the molecular weight M may be 
calculated on the basis of Avogadro's law according to the 
relation (p. 19) 


G : 24-45 ~ T - = M: 2, 

where G is the weight in grams (= i), V the volume in 
litres (01006), so that 

*= 0.813 x. 

The numbers thus calculated are given in the last 
column, and are in good accord with the number, 342, 
obtained from the formula of sugar, C l2 H 29 O n . Notwith- 
standing this and several other researches l , direct measure- 
ment of osmotic pressure has not yet led to a practicable 
method for determining molecular weights. 

2. Indirect Methods for Molecular Weight Determination. 

Whilst direct measurement of the osmotic pressure has 
so far met with difficulties for want of a satisfactorily 
resistent semipermeable membrane, so that the experiment 
is difficult or even impossible, attention has been paid to 
other properties of solutions, which bear calculable relations 
to the osmotic pressure. 

With regard to these methods, we may say generally, 
with Nernst, that every mode of separation between solvent 
and dissolved body involves a method for determination of 
molecular weight. This is so because every such separation 
renders possible a cyclic process in which the solvent, after 
having been separated in the way in question, may be again 
brought together with the separated or dissolved substance 
osmotically, i. e. by means of a semipermeable membrane. 
If this cyclic process be carried out reversibly e. g. if the 
osmotic action be supposed to take place in a cylinder with 
piston, such that the wall of the cylinder enclosing the 

1 Ladenburg, BerL Ber. 22. 1225 ; Adie, .7. C. S. 1891, p. 344 ; Ponsot, 
Compt. Rend. 125. 867 ; Naccari, Rend. Ace. Lin-cei, 1897, i. 32. 


solution is semipermeable, and the piston exercises a 
pressure exactly equal to the osmotic pressure then 
a thermo-dynamic relation may be deduced which involves 
the possibility of measuring the molecular weight. 

Viewed from this point of view, the indirect methods for 
determination of molecular weights may be divided into 
two groups, according as the cyclic process underlying them 
can be carried out at constant temperature or not; in 
the first case the theoretical argument is simpler, and can 
often be simply demonstrated without the aid of thermo- 

(a) The cyclic process can be carried out at constant 
temperature. Molecular weight determination by vapour 
pressure measurement. The common character of methods 
based on cyclic processes which can be carried out at 
constant temperature is that they can be performed at any 
temperature, whether the separation be as vapour or by 
shaking with a third substance. We have, however, to 
distinguish as distinct methods the separation of solvent 
and of dissolved substance, and so we arrive at the following 
resume' : 

1 . Separation as vapour. 

(a) The dissolved substance separates out : phenomena 

of absorption. 
(/3) The solvent separates out : lowering of vapour 


2. Separation by a solvent. 

(a) The dissolved substance separates out : ratio of 


(/3) The solvent separates out : lowering of solubility. 
With regard to the first of these methods, based upon the 
phenomena of absorption, we may refer to p. 28, and only 
note that when Henry's law of absorption holds (propor- 
tionality between pressure of the gas and concentration in 
solution) the absorbed gas in question must exist with 
unchanged molecular weight in solution. 

We must now deal with the second method, based on the 



lowering of vapour pressure produced by the dissolved 

The vapour pressure of a solution is immediately related 
to its osmotic pressure. Obviously two solutions in the 
same solvent, if they have the same osmotic pressure, i. e. 
are isotonic, must show the same 
saturation pressure. Otherwise a per- 
petuum mobile in the meaning of 
Fig. 7 would be inevitable. The two 
isotonic solutions A and B are sup- 
posed separated by a semipermeable 
membrane ; then if there is a difference 
between their saturation vapour pres- 
sures, a current of vapour will flow 
in the upper part of the vessel, say 

from left to right. Then the decrease and increase of 
concentration produced in B and A respectively will cause 
a movement of the solvent through a b from right to left, 
and so the conditions for a perpetual motion are given, 
which can only be avoided if isotony goes with equality 
of vapour pressure. 

But further, the magnitude of the change in vapour 
pressure which a solvent suffers in taking up a dissolved 
body may be calculated, as will be clear from what 

Deduction of the law of diminution of vapour pressure 
without thermo-dynamics. We will first give the simple 
proof, due to Arrhenius, which does not 
involve thermo-dynamics. Suppose os- 
motic equilibrium to be established by 
means of the semipermeable membrane 
a b (Fig. 8), above which the solution rises 
to the height AB. Now the equilibrium 
implies that the solvent should not pass 
by distillation into or out of the solution. 
But the pressure, which at B equals the saturation pressure p 
of the solvent, is less at A by the pressure exerted by the 


column of vapour AB ; accordingly, the saturation pressure 
of the solution which obtains at A must be less than that of 
the solvent, by say A^>. But just as the diminution 
of vapour pressure corresponds to the column of vapour, 
so the osmotic pressure corresponds to the column of liquid 
resting on a b, and if both be referred to the same area, the 
change of vapour pressure is to the osmotic pressure as 
the weight of any volume of vapour is to that of the same 
volume of liquid. Taking, for simplicity, a kilogram- 
molecule M of vapour, if its volume is v litres, then by 
applying Avogadro's law to the vapour and solution we get 

pv = PV, so that v V, 

where p and P are respectively the vapour pressure and 
the osmotic pressure, and V is the volume in litres of 
a kilogram-molecule of the dissolved substance. If the 
composition of the solution is given as n molecules of 
dissolved substance to N molecules of solvent, one dissolved 

molecule ??i occurs in in + M kilograms of solution (here 

the molecular weight of the solvent is taken as M, as 
determined from the vapour density, but without any 
assumption as to the real value of that quantity in the 
liquid). If N is great, as in dilute solutions, the weight 


may be taken as - ; this is accordingly the weight of the 

volume V of solution, and that of v is - - 

p n 

Hence we have the relation 

PNM ,, 

P : A p = ~ - :M 


A n 

This is a somewhat modified form of Raoult's law. The 
latter refers to the so-called molecular lowering of vapour 


pressure, i.e. the value of /\p/p obtained for a one percent, 
solution multiplied by the molecular weight of the dissolved 

substance, or 

Ap __ 7i 

<pT' 'K 11 

in which one part of dissolved substance is contained in 
i oo parts of solvent : 

nm : NM = i : 100 or-^ = 1-01 M, 


m = ooi M: 


in other words, the relative lowering of the vapour pressure 
is equal to one hundredth of the molecular weight of the 
solvent ; in which it must be borne in mind that the latter 
quantity has the value derived from vapour density 

The following table shows results obtained for moderately 
dilute solutions * : 

Solvent M m 


Water 18 0-185 

Phosphorus trichloride 137*5 i'49 

Carbon disulphide 76 0-8 

Carbon tetrachloride 154 1-62 

Chloroform 119-5 i-3 

Amylene 70 0-74 

Benzene 78 0-83 

Methyl iodide 142 1-49 

Methyl bromide 109 1-18 

Ether 74 0-71 

Acetone 58 0-59 

Methyl alcohol 32 0-33 

Mercury 2 200 2- 

It may be remarked that, as with all rules derived from 
the theory of solutions, it is only strictly true for infinite 
dilution. For that reason we will now repeat the proof 

1 Raoult, Compt. Rend. 87. 167. 

2 Ramsay, Zeitschr. f. Phys. Chem. 3. 359. 


with the aid of thermo-dynamics, insisting again that the 
law of the lowering of vapour pressure infers nothing as 
to the molecular weight of the liquid solvent. 

Thermo-dynamic deduction of the laiv of diminution of 
vapour pressure. Let the solution contain da kilograms of 
dissolved substance in one kilogram of solvent, the former 
possessing the molecular weight m; withdraw from it, 
osmotically and reversibly, i. e. with the aid of cylinder and 
piston, so much solvent as contains one kilogram-molecule 
of dissolved substance, with expenditure of work 

where dP is the osmotic pressure in kilograms per square 
metre produced by the absorption into the solution of da 
per kilogram, and Fthe volume in cubic metres in which 
one kilogram-molecule is dissolved. 

The amount of solvent withdrawn is to be restored 
reversibly by evaporation and condensation. In the first 
place we gain a quantity of work (i) by reversible evapora- 
tion at p and T ; next a second quantity by expansion of 
the vapour till the pressure has fallen to p dp, that of the 
solution. This work, for the kilogram-molecule M of 
vapour, is 

Avdp = iT^; 


consequently, for the mass -7- with which we are concerned, 


2 Tdp m 
p Mda 

Finally the vapour is to be condensed in contact with the 
solution at p dp and T, in which process the work gained 
in (i) is used up. Since this cyclic process has been carried 
out at constant temperature, it cannot be accompanied by 
a conversion of work into heat, or vice versa, so that the 


osmotic work spent must be equal to the work of expansion 
gained, i. e. 

zTdp m 

* T = TO- ' 
p Mda 


m = Mda, 

which again comes to Eaoult's law, if we remember that 
the solution contains da to one or icoda per cent. Accord- 
ingly, for a one per cent, solution, assuming the proportion- 
ality between concentration and lowering of vapour pressure 
which is true for infinite dilution to hold, we have 

Ap dp in 
^-m = - i 

p p looda 

In the same way as before we arrive at the expression 
&p n 

~^ := F ' 

The meaning of this equation may be illustrated by 
answering the question, ' What fall of vapour pressure does 
i / o of sugar produce in water at 100 ? ' Here p = 760 mm. 

i 100 i 

n : 

-- -- - 

342 i 8 1900 

so that 


- = 

Abnormal values for abnormal vapour densities. There 
is another essential point to be noticed, viz. that N, 
the so-called number of molecules of solvent used in the 
deduction, is not always the real number ; rather N is the 
fictive number of molecules obtained by taking the mole- 
cular weight of the saturated vapour at the temperature 
considered as unity. If the vapour, then, has an abnormal 
density, the abnormal molecular weight so obtained is to 


be used in calculating N, as Raoult and Recoura l showed 
to be the case for acetic acid. They measured, as explained 
above, the relative molecular lowering of vapour pressure, 
i.e. the lowering for a one per cent, solution multiplied by the 
molecular weight of the dissolved substance, or in Raoult 's 

/-/' TO> 

where P is the percentage content of solution, / /' the 
lowering of vapour pressure. This molecular diminution, 
multiplied by 100 and divided by the molecular weight M 
of the solvent, gives a number that averages unity, or 

f-f m_ 

We get the same result from the formula already given : 

= ooi M* 

w^hich coincides with the former if only p, the pressure of 
the solvent, is put for /', that of the solution, which is 
permissible in dealing with very dilute solutions. 

On the other hand, acetic acid gave 1-61 instead of I 
when Jl/was taken as 60, according to the formula C V H 4 2 . 
But to do so would be to choose too small a molecular 
weight, for that, according to the vapour density, is 97. 
Taking the correct value we again get a number close to 
unity, as before, since 

i-oi x =i. 


Hence, in determining the lowering of vapour pressure 
lies another means of measuring the molecular weight of 
the saturated vapour of the solvent, provided the molecular 
weight of the dissolved body be known. 

1 Zeitschr.f. Phys. Chem. 5. 423 ; Berl. Ber. 29. Ref. 941. 



Accuracy attainable in measuring vapour pressures. 
Finally, something may be said as to carrying out the 
measurement of vapour pressure, and the accuracy attain- 
able in doing so. As compared with the method of 
measuring the osmotic pressure directly, the vapour pressure 
method stands at a considerable disadvantage. The one per 
cent, sugar solution, that at ordinary temperatures exercises 
f atmosphere osmotic pressure, causes a fall of only 0-4 mm, 
in the vapour pressure at 100. But on the other hand, 
the measurement of osmotic pressure has met with such diffi- 
culties in practice, that several attempts 
have been made recently to obtain more 
accurate results by the vapour pressure 
method 1 . An important improvement 
lies in increasing the difference of level 
which corresponds to the pressure differ- 
ence to be measured. The substitution 
of oil for mercury (Frowein-Bremer 
tensimeter 2 ) is a great advantage. But 
lately Smits has applied a suggestion of 
Kretz (Jamin, Cours de Physique, III, 
vol. iv, p. 218) in using two liquids 
differing but little in density. One the 
heavier was aniline, which, when moist, has at 20 the 

s a = 1-022; 
the other was water, whose density at 20 is 

8 b = 0-998. 

The heavier liquid occupies the lower part of the narrow 
tube AB (Fig. 9) ; the water, for the most part, the reservoirs 
c and D (and is covered with a thin layer of oil, not shown 
in the diagram, to prevent evaporation). Then suppose an 

1 Dieterici, Wied. Ann. 62. 617 ; Smits, Arch. Neerl 1897 ; Boldingh, 
MaandLl. voor Natuurw. 21. 181. 

2 Zeitschr.f. Phys. Chem. i. 424. 

Fig. 9. 


increase of pressure of p mm. of mercury applied on the 
right-hand side, and let h be the difference in level of 
the aniline produced, a the ratio of the cross section of the 

reservoir to that of the tube, so that - is the difference in 


level produced in the reservoirs ; then 

13-56 > = k (1-022 0-998) H 0^998, 


so that if a is very large 


^ = 5 6 5P> 

i. e. in the most favourable case i mm. of mercury will 
make 565 mm. difference in level. 

With such apparatus the following result was obtained 
for cane sugar (C 12 H 22 O n ) : 

7-31198 grams in loco grams of water gave at o 6 , for 
which p = 4-62 mm,, a value of A_p = 0-00178 mm. 

If we determine the molecular weight of sugar from 
these data we get : 


















7-31198 x 18 






= 342 

in exact agreement with the formula C 12 H 22 O n . 

Molecular weight determination by lowering of solu- 
bility. As previously remarked, every means of separation 
between solvent and dissolved body implies a possible 
measurement of molecular weight. If the separation is 
effected by evaporation, then if the solvent alone is volatile 
we arrive at the law already given ; if on the other hand 
only the dissolved body is volatile i. e. a gas then 


a decision as to its molecular weight may be based on 
Henry's law. But the separation may also be accomplished 
by the addition of a liquid which does not mix with the 
solution. Again two methods arise, according as we deal 
with the solvent or dissolved body passing over into the 
new liquid. In the first case we may expect the analogue 
of the law of diminution of vapour pressure, in the second 
that of Henry's law. 

Let us take first the lowering of solubility, which was 
first applied to determine molecular weights by 
Nernst l , and afterwards carried out in a very 
simple manner by Tolloczko 2 . The whole 
measurement of molecular weight could be carried 
out in a flask (Fig. i o) of volume about 1 80 c.c. with 
a long narrow neck (i cm. = 0-4385 c.c. ; content 
13 c.c.) divided in half millimetres. As liquids, 
ether and water were chosen, and the decrease of 
solubility of ether in water brought about by Fig. 10. 
any substance soluble in ether measured. 

To take an experiment in detail: a known quantity of 
water saturated with ether at 18 -3 was used, containing, 
according to a previous measurement of solubility, 10763 
grams ether. This quantity of ether saturated with water 
would occupy 34-967 scale divisions of the neck. The ethereal 
layer occupied 4-86 scale divisions, and so weighed 1-4958 
grams ; on addition of 0-0952 gram benzene it increased to 
6-1 1, equivalent to 1-88 grams. 

As previously for the vapour pressure we get the 

in which p and Ap are the pressure and decrease of 
pressure respectively, n and N the number of dissolved 
and dissolving molecules (the latter calculated from the 

1 Zeitschr. f. Phys. Chem. 6. i. 

2 1. c. 20. 389. 

D 2 


molecular weight obtaining in the saturated vapour), so 
now we have 

As 7i 

where s and As are the solubility and decrease of solubility 
respectively, n and N the number of dissolved and dissolv- 
ing molecules (the latter calculated from the molecular 
weight obtaining in saturated solution, here therefore 74, 
the molecular weight that ether gives in saturated aqueous 
solution). Thus 

As : s = (6-1 1 4-86) : 34-967 = 0-0358 

0-00^2 1-88 

n:N= ^~ : -. ^ ^ ^ = 0.0358. 
x 7 4 (=C 4 H 10 0) 

The value of x (molecular weight of benzene) so found 
would be nearer to the expected value if we took into 
account the solubility of water in ether, and the alteration 
in that solubility brought about by addition of benzene. 
Thus at the temperature of the experiment some 1 1 molecules 
of water are dissolved in 100 of ether. 

On this account the so-called molecular displacement 

m = 


should be determined by means of a body of known mole- 
cular weight, and for a given mass of water and tempera- 
ture, g being the mass of substance dissolved always in the 
same mass of ether. Then a molecular weight desired may 
be deduced from the value of C. 

Example : 0-0655 gram of benzene effected a displacement 
of 0-45 cm.; the molecular weight of benzene is 78 (C 6 H G ), 
so that 

Then 0-1266 gram of naphthalene was taken, giving the 


displacement 0-55 cm., and hence for the molecular weight 
of naphthalene 

tv6 = xx L -^- ? 3 

x 123 

which for practical purposes agrees sufficiently with the 
formula C 10 H 8 = 128. 

Ratio of partition. As remarked on p. 43 a measure- 
ment of molecular weight may be based on the ratio of 
partition, and the law with regard to this may be most 
simply arrived at by considering the equilibrium of the 
vapour of the dissolved body, the mutual solubility of 
the two liquids being neglected. If the molecular weights 
in the vapour and in the two liquid phases be m, (m) /v and 
(m) ; , 2 respectively, and the concentrations c, c v c 2 , then on the 
one hand 

and on the other 

= & 2 , 


so that 

l - = -^- = i and (w~) = (77) ^ 

where c 1? c 2 , (7 1? (7 2 are the concentrations as found in two 

Hence if n^ is known, i. e. the molecular magnitude in 
one liquid, then two measurements of partition will give n. 2 , 
the magnitude in the other solvent. 

As a rule the molecular magnitude is the same in both 
solvents, being simple in each, and so we have 

?&! = n 2 and - = k, 

i. e. constant ratio of partition, as found in certain cases by 
Berthelot and Jungfleisch 1 . But as hydroxylic substances 

1 Ann. de Chim. et de Phys. (4) 26. 396, 408. 


in hydroxyl-free solvents mostly occur with double mole- 
cular magnitude (p. 60) an inconstant ratio of partition is 
found, e. g. for acetic acid in water and benzene l : 


(gm. acetic acid 

(gm. acetic acid 



in 31-5 gm. benzene) 

in 5-075 gm. water) 



















The fair agreement of the values of - =- as compared with 


the decided falling off in the values of agrees with the 

c i 
conclusion arrived at also from other sources, that acetic 

acid in water exists as C. ? H 4 O 2 , but in benzene practically 
as (C 2 H 4 Cg 2 . 

(b) The Cyclic Process cannot be carried out at Constant 

Whilst in the foregoing methods the underlying cyclic 
process can be carried out at constant temperature, we 
have now to consider methods in which that is not 
the case. 

Molecular weight determination by lowering of the freez- 
ing point. Separation of the solvent in the solid state, 
i. e. freezing of the solution, affords, as is well known, the 
oldest method for molecular weight measurement, which 
was developed on an empirical basis by Raoult before the 
theory of solutions was established. 

The cyclic process which again leads to the rule to be 
applied, cannot be carried out at constant temperature, and 
so involves the use of the second law of thermo-dynamics. 
The solution is to be cooled by A t below the freezing point 
of the solvent (T absolute), and at that temperature 
a certain quantity of ice separated, a process which can be 

1 Nernst, Zeitschr.f. Phys. Chem. 6. 121. 


effected at the constant temperature TAt if the mass of 
solution is so great that the separation in question makes 
but an inappreciable change in the concentration. Then the 
solution and solid solvent are to be separated and warmed 
to T, the solvent melted and mixed with the solution in 
an osmotic reversible manner (with cylinder, piston, and 
semipermeable membrane), a certain amount of work being 
gained. If A t is very small (df) a quantity of heat Q has 
in the cyclic process fallen in temperature by A, which 
implies a production of work 

If the solution contains a per cent, and so much of ,the 
solvent is frozen out as contains one kilogram-molecule m of 

, , , . loom ., 
the dissolved body, i. e. - ? then 

where W is the latent heat of fusion of ttie solvent 
per kilogram. The work gained is, according t to the 
previous argument, 

so that 

.a T 


A t 0-02 T 2 

m = 

- ^ 

a W 

Here, however, the first term is the so-called molecular 
lowering of the freezing point f, i.e. the lowering for 

a one per cent, solution , multiplied by the molecular 


weight m. 

Thus for water, from T = 273 and W = 80 we may 

t= 18-6 


(18-5 if instead of zT the more exact value I-98T is 
taken), so that a one per cent, solution of methyl alcohol 
(CH 3 OH = 32) would show a melting point # = 0-58 
below zero, since 

xx 32=18-6 or x = 058. 

Vice versa, from a known depression of the freezing 
point the molecular weight may be calculated, and since in 
determining molecular weights the choice always rests 
between values differing widely from one another, un- 
ambiguous results may be obtained by the use of the 
process worked out by Beckmann and Eykman. If, how- 
ever, the object is to test strictly the formula deduced 
above, measurement of the freezing point is found to be 
not altogether easy. Sugar again has been the subject of 
many experiments, and the following table shows that only 
of late years has satisfactory agreement been reached : 

1885 Kaoult 1 .... 18-5 (i per cent.) 

1886 Raoult 2 .... 25-9 (very dilute) 
1888 Arrhenius 3 .... 20-4 (i^ per cent.) 
1893 Loomis* .... 17-1 (% per cent.) 

1893 Jones 5 ..... 23-7 (p. 08 per cent.) 

1894 Ponsot 6 .... 18-77 (infinitely dilute) 

1896 Abegg 7 ..... 18.6 (infinitely dilute) 

1897 Wildermann 8 . . . 18.7 (0-17 per cent.) 
1897 Raoult 9 .... 18-72 (very dilute) 

It is especially noteworthy that Raoult, in 1897, withdrew 
the opinion expressed by him in 1885 in favour of abnor- 
mally great depression for sugar in very dilute solution, 
and so fell into line with the theory of solutions. 

It should always be remembered that the formula 

A 0-02 T 2 

m = ^f 

1 Compt. Rend. 94. 1517. 2 Ann. de Chim. et de Phys. (6) 8. 313. 

3 Zeitschr.f. Phys. Chem. 2. 497. 4 Berl. Ber. 26. 800. 

5 Zeitschr.f. Phys. Chem. 12. 642. 6 Compt. Rend. 118. 977. 

7 Zeitschr.f. Phys. Chem. 20. 230. 8 Journ. Chem. Soc. Trans. 1897, p. 796. 

9 Ann. de Chim. et de Phys. (7) 10. 79. 


is only strictly true for infinite dilution, so that the form 

dt _ 003 T 2 O'OiqST 2 

~ i/i/ TIT " or TT -r- 

da W W 

is to be preferred, where da is the dissolved substance in 
100 parts. 

Molecular ^veight determination by the rise of boiling 
point. Just as the solvent can be removed by freezing, so it 
can be separated from the dissolved body by boiling, and the 
lowering of vapour pressure already dealt with corresponds 
to a rise in the boiling point. This may be determined in 
magnitude by means of a cyclic process similar to that 
referring to the freezing point, and amounts to 

0-02 T- 00108 T 2 
- ir - or exactly- -^ , 

where t is the molecular rise of boiling point (i. e. -7- m, 


p. 56), T the absolute temperature of the boiling point, 
W the latent heat of evaporation. The method of molecular 
weight determinations ba"sed on this has also been worked 
out by Beckmann. It is usually not so good as the freezing- 
point method, because the molecular rise of boiling point 
is, on account of the great latent heat of evaporation, 
relatively small. 

This will be plain from the following table, which gives 
the freezing point (Fp), latent heat of fusion (TF}) 3 boil- 
ing point (Sp), and latent heat of evaporation at the 
boiling point ( TT 8 ), together with the molecular change in 
freezing point (Mf), and boiling point (M 8 }> calculated from 

the formula 

#\& OF TH 
,_ 0-0198 T 2 




W f 

M f 


w s 

M 8 

Water . . . 







Acetic acid . . 







Benzene . . . 








It may be added that recently the boiling-point method 
lias been applied in a simpler form by Landsberger l and 
Walker 2 , by passing e. g. ether vapour, generated in a 
flask, through the ethereal solution of the substance investi- 
gated, till the thermometer no longer rises, and indicates 
the boiling point of the solution. 

Molecular y:eight determination by means of the change 
of solubility ivith temperature*. If no third substance is 
employed to effect the separation between dissolved body 
and solvent, and we omit that in the list of methods, it is 
in the first place difference in state of aggregation which 
allows of the separation. Thus molecular weight determina- 
tion by means of vapour pressure and boiling point is 
possible on account of the separation of solvent as vapour ; 
determination by means of the absorption coefficient is 
possible on account of the separation of the dissolved body 
as gas. Separation of the solvent in the solid form leads 
to the method of the lowering of the freezing point, and 
finally the account may be completed with the possibility 
of separating the dissolved body in the solid form. The 
method so indicated may now be taken. 

The method is similar to that applied on p. 21 to gas and 
vapours, involving the well-known formula 


Here V is the increase of volume due to evaporation of 
a definite quantity, and g, the heat absorbed, refers to the 

With regard .to solutions this equation may be trans- 
formed, according to Part I, p. 36, into 

d( Q 

dT 2T* CdT~~ 2^' 

1 Berl. Ber. 31. 458. 2 Journ. Chem. Soc. 73. 502. 

3 Verschaffelt, f. Phys. Chem. 15. 449 ; Van Laar, 1. c. 15. 473 ; 
17. 545 ; 27. 337 ; Goldschmidt, 1. c. 17. 145 ; 25. 91 ; Noyes, 1. c. 26. 699. 


in which Q is the heat absorbed on solution of one kilogram- 
molecule, C the concentration. 

This formula allows of a molecular weight determination 
of the dissolved substance, by calculating L from the data 
on solubility, and dividing by the observed heat of solution 
per kilogram ; thus for succinic acid (C 4 H 6 O 4 ) the amounts 
dissolved in 100 parts of water at o and 8-5 respectively 
are 2-88 and 4-22 ; hence we get from 

A f _Q_ 

_ _ _ 

~- . = 

The heat of solution per kilogram is 55, and therefore the 
molecular weight =124 (C 4 H 6 4 = 118). 

C. Besults. 

i. Simple Molecular Magnitude of Dissolved Bodies. 

Putting together the results of the numerous determina- 
tions of the molecular weights of dissolved bodies, the 
leading conclusion arrived at appears to be that the mole- 
cular magnitude is usually that corresponding to the 
simplest formula that answers to the quantitative composi- 
tion, and the chemical relations of formation and transforma- 
tion. This conclusion has helped not a little towards the 
favourable reception of the theory of solutions, and to 
the working out of the molecular weight methods based 
on it. 

Larger molecules. Agreement ivith the results of gas 
density measurements. In the first place, departures from 
the simplest possible molecular formula are found in cases 
in which the gas density indicates the same thing. The 
molecular magnitude of the elements, already referred to, 
may be mentioned in this regard. Most are diatomic : so 
also are oxygen, nitrogen, and hydrogen in aqueous solution, 


since Henry's law of absorption applies to them (p. 31) ; 
for iodine the same has been shown by measurements of 
the freezing point ' . Again, the monatomicity of the metals 
(p. 35), deduced from their vapour densities, is found to 
be true for solution in mercury 2 , whilst phosphorus and 
sulphur appear as tetr- and oct-atomic respectively, as in 
the form of vapour. 

Kesearches on the molecular weights of the elements can 
thus be carried much further, since the non- volatility of the 
metals is here of no consequence, and so nearly all the 
metals have been studied in solution in tin, and have mostly 
proved monatomic. 

As to compounds, the tendency of acetic and formic 
acids to form double molecules, shown by the vapour 
density, recurs in many (hydroxyl-free) solvents 3 . 

Larger molecules in hydroxylic compounds. What does 
not appear from the investigation of gases and vapours is 
the tendency of nearly all hydroxylic compounds to form 
double molecules when in somewhat concentrated solution, 
as was mentioned for formic and acetic acids. This is found 
to be the case in general for organic acids, for the alcohols, 
and for water. The presence of double molecules, however, 
depends on the solvent, being found only in hydroxyl-free 
liquids, such as the hydrocarbons, chloroform, and carbon 
disulphide. In hydroxylic solvents, phenol, acetic acid, 
water, the reduplication does not appear, either because 
of a dissociating action of the solvent, or because the dis- 
solved body forms molecular complexes with the solvent, 
in which only one molecule of dissolved occurs. Hence in 
practice, as the smallest molecular weight answering to the 
chemical behaviour is sought, it appears that hydroxylic 
solvents such as acetic acid are to be preferred, especially 
when dealing with hydroxylic bodies. 

1 Beckmann, Zeitschr. f. Phys. Chem. 5. 76 ; 17. 107 ; 22. 614 ; Helff, 1. c. 
12. 219; Aronstein and Meihuizen, Verh. Kon. Akad. Amsterdam, 1898. 
3 Ramsay, Zeitschr. f. Phys. Chem. 3. 359 ; Tammann, 1. c. 3. 441. 
3 Beckmann, 1. c. 2. 742. 


2. Development of the Stereochemical Conceptions. 

Molecular weight measurements play a specially important 
part in studying the cause of the isomerism between two 
bodies of the same composition. For it is then possible 
to decide whether there is a difference in molecular magni- 
tude, i. e. polymerism, or a difference in construction between 
two molecules of the same size, i.e. isomerism in the 
narrower sense. It may be remarked here, that with 
regard to the development of stereo-isomerism, which is 
a special kind of isomerism attributed to the arrangement 
of the molecule in space, molecular weight determinations 
form an indispensable aid, since it is necessary first to 
prove that differences between equally composed molecules 
are in question. Hence it was 'a fortunate coincidence that 
just at the time the theory of stereochemistry was dis- 
covered, the new methods for the measurement of molecular 
weights in solution, based on the theory of solutions, were 
introduced. The interesting cases of isomerism in the 
truxillic acids, the benzoinoximes, and many other com- 
pounds important for stereochemistry, could hardly have 
been so quickly and certainly brought into light without 
the new methods for molecular weight determination. 

3. Abnormal Results for Isomorphous Compounds. 

A very striking exception appears when a substance is 
dissolved in another isomorphous with it, as thiophene in 
benzene. The lowering of the freezing point is then 
decidedly less ; in the case mentioned only about two-thirds 
of the normal amount. In such cases, as has actually been 
shown 1 , the dissolved substance separates out in an iso- 
morphous mixture with the solvent when the latter 

1 Van Bylert, Zeitschr. f. Pliys. CJiem. 8. 343; Beckmann, I.e. 17. 107; 
22. 609. 


freezes ; hence we have a departure from the assumptions 
made in determining the formula 

O-O2 T 2 


and it will appear later that for that reason the lowering 
of the freezing point should be smaller. 

The phenomenon in question has been applied by 
Ciamician and Garelli 1 to settle doubtful cases of iso- 
morphism, and by that means the constitutional formula, 
since it is known that isomorphism goes hand in hand 
with similarity of constitution. 

4. Abnormal Results for Electrolytes. 

Lastly we must consider the great exceptions which 
occur in electrolytes, i.e. especially solutions of salts, 
strong acids, and bases in water, where the lowering of 
the freezing point often amounts to twice or more times 
the normal amount 2 . It was just this phenomenon that 
so long stood in the way of the discovery of the simple 
laws applicable to dilute solutions, which first came to 
light when Raoult made his freezing-point measurements 
on non-electrolytic solutions, i.e. on solutions in solvents 
other than water, and solutions in water of organic 
substances not of a salt character. It is well known 
that the phenomenon in question led to the theory of 
electrolytic dissociation, which now requires special 

The Theory of Electrolytic Dissociation. Arrhenius 
attributed the abnormally large change of freezing point 
just mentioned, and the abnormally high osmotic pressure 
deduced from it, to a decomposition into ions, i. e. electri- 
cally charged portions <af molecules. The extensive data 

1 Zeitschr.f. Phys. Chem. 13. i, 18. 51, 21. i. 

2 The greatest deviation has lately been found by Crum Brown for the 
salts of sex-basic mellitic acid. 


accumulated in support of this view will only be partly 
mentioned here. In the Third Part of these lectures, in 
which the relations between constitution and properties 
are to be discussed, many opportunities will arise to draw 
qualitative and quantitative conclusions from the idea of 
Arrhenius, as was the case in the First Part (p. 117), in 
dealing with the conditions of equilibrium in electrolytes. 
Here we are specially concerned with the question, on 
which the decisive proof turns, whether the degree of 
decomposition deduced from the theory of solutions agrees 
with that arrived at independently on the basis of the 
electrolytic dissociation theory. 

The form in which the departure of electrolytes from 
Avogadro's law, as applied to solutions, is to be given, may 
conveniently follow on the so-called isotonic coefficients 
introduced by de Vries (p. 38) ; these give how many 
times more effective a molecule of salt in an isotonic 
solution is, with regard to plasmolytic or osmotic action, 
than a sugar molecule, the latter being, for special reasons, 
taken as = 2. We have modified this mode of expression 
by taking sugar and other substances following Avogadro's 
law as unity, and writing the halved isotonic coefficients 
so arrived at with the letter t. If then, for a certain con- 
centration, i = 1-75 for nitre, that means that if each molecule 
of nitre were replaced by 1-75 molecules of sugar, a solution 
isotonic with that of nitre would be obtained. Hence the 
osmotic pressure of the nitre solution is 1-75 times the 
normal amount, so that the quantity t may also be 
obtained as the ratio between the molecular depression 
of the freezing point t proportional to the osmotic pressure, 
and the normal value i8'5 found for sugar and other 
non-electrolytes, or 

For the same temperature and concentration, the same 
value of t should be obtained, whichever of the methods 


already mentioned is adopted whether based on the law 
of absorption, on the freezing point, or on direct measure- 
ment of osmotic pressure; and the confirmation that 
Arrhenius' conception has received lies in the agreement 
between the value of i so obtained and that independently 
reached on the basis of that conception. 

In the first place then we must consider the relation 
between i and the conductivity. 

Electrolytes have, it is known, a conductivity which, 
referred always to the same say normal concentration, 
increases with increasing dilution. Calling this the mole- 
cular conductivity ^ the conductivity of mercury at o 
being taken as unity, and the number multiplied by io 7 
for convenience we have for KNO 3 at 1 8, according to 
Kohlrausch : 

Normality - 

A* = 




























120 7 




i.e. a number which increases asymptotically as the dilution 
becomes greater. The explanation of this fact on Arrhenius' 
theory is that with the dilution comes an increasing ionic 
decomposition according to the equation 

KN0 3 = (K) + (N0 3 ), 


where (K) and (N0 3 ) represent the positively and negatively 
charged ions respectively. This decomposition is practically 
complete for the limiting value 

Theoretically it would only be reached on infinite dilution, 
and is therefore represented by /z^. 

If the conductivity depends, as we have assumed, only on 
the ions present, the quotient 


represents the fraction decomposed, and the total number 
of molecules, counting each ion as such, exceeds the 
normal amount in the ratio 

I : (T a) + 2 a I : I + or, 
so that we have 

i = i+a= I+ 

Comparing a special case directly with de Vries' experi- 
ments in which urea was chosen as normal substance, we 
find for the concentrations mentioned the following results 
(plasmolysis p, no plasmolysis n, intermediate state np) : 

Normality Urea Nitre 

= 0-285 0-3 0-315 0-33 0-345 

n n np p p 

N = 0-16 0-17 0-18 0-19 0-20 

n n np p p 

so that 0-315 normal urea is isotonic with 018 normal 
nitre. Accordingly for a 018 normal nitre solution 


' = -StlF = '** 

By the conductivity we get, interpolating for the same 

jui = 99-7, so that i i+ = 14- ^^ = 1-81. 

Moo 1 '22, 

The agreement is here satisfactory. Unfortunately it has 
not in all cases remained equally so on application of the 
more exact procedure of the freezing point. 

It would be very desirable to form a collection of results 
obtained by the freezing-point method, to serve as a test 
of Arrhenius' law by comparison with the collection of data 
011 conductivity 1 . 

Under these circumstances it seems desirable to choose 
Ostwald's ' law of dilution ' as a second characteristic, and 
divide electrolytes into two groups according as they follow 
it or not. 

1 Das Leiti-ermogen der Elektrolyte, inslesondere der Losungen, Kohlrausch and 
Holbom. Leipzig, 1898. 



Electrolytes luhicTi follow Ostivald's law of dilution. For 
a dissociation in a dissolved electrolyte in the sense of the 

KN0 3 = (K) + (N0 3 ), 
i. e. an equilibrium symbolized by 

if the ions are treated as molecules it seems an inevitable 
consequence of the equation of equilibrium already ar- 
rived at, 

2 n log C const., 

the concentration of the two systems being taken with 
opposite signs ; this may be written 

or if V be the volume in litres of one gram-molecule (reci- 
procal of the normality), 

~ I a ^ p + a n 


(i a) V MooO^oo P)'* 

However, for salts, e. g. nitre, this relation does not hold, 
as we shall see later, but only for a definite though very 
numerous group of electrolytes, viz. acids and bases in 
aqueous solution, that decompose in the manner 

ZH = (Z) + (iff) and MOH = (M) + (OH) 9 

and of these only the weak acids and bases. The law 
seems to be restricted to cases in which the dissociation 
yields ions of the solvent water, and then only when their 
concentration is small. 



As an example, the data for chloracetic acid at 14 are 
as follows l , the calculated values of - - being derived from 

the equation 

M 2 
7 ' r^r = " 

















?ogr^T=7'2 10 










Clearly here the decomposition as measured electrically 
follows the course anticipated on the theory of dissociation. 
Accordingly we find, e. g. for dichloracetic acid, which also 
follows Ostwald's law, that the values of t obtained from 
the molecular depression of the freezing point (M), and 
those from the conductivity, agree 2 . 









The agreement is satisfactory, and might well be complete 
if we remember that the freezing-point measurement 
always the more difficult refers to o, while the electrical 
is for 1 8. 

Electrolytes which do not follow Ostwald's law of dilution. 
The case is quite different for salts and strong acids and 

1 Van 't Hoff and Reicher, Zeitschr.f. Phys. Chem. 2. 781. 

2 Wildermann, 1. c. 19. 242. 

E 2 


bases. If, e. g. 3 we take the former value of the conductivity 
of nitre, and calculate 

log ; r-r^j 

II I 11 - 11 I w 

where V is the reciprocal of the normality and ^ = 122, we 
get : 



= 3 

= 572 


I - 

75-2 839 
2.081 1-9657 












1-5864 I 






122 H 






II9. 9 



120 7 


122 a 


The relation suggested by Rudolphi and van't Hoff' 1 , 

" 3 rr M 

-^= = K, a = , 

does not remove this difficulty in principle, for though 
it certainly agrees better with the facts (Part I, p. 124) it 
has no necessary basis, and so can at present only be 
regarded as empirical. 

Moreover, there is a considerable difference between the 
values found for the dissociated part 2 (in per cent.), e. g. of 
sodium chloride, by the freezing-point method by Jones, 
Loomis, Abegg, Kaoult, and from the conductivity by 
Kohlrausch : 























































1 Zeitschr.f. Phys. Chem. 17. 385, 18. 305 ; Kohlrausch, 1. c. 18. 661. 

2 Noyes, 1. c. 26. 709. 


Evidently, then, dissolved salts, strong acids and bases, 
must he treated with caution as electrolytes, for though 
there is undoubtedly dissociation, and the assumption of 
dissociation into ions appears to be indispensable, yet the 
exact amount of that dissociation, and the mechanism of it, 
are insufficiently explained. In these cases it is necessary 
to hold by the thermo-dyiiamic relations based on vapour 
and osmotic pressures, &c., and to treat the calculations 
made from conductivity data as rough estimates, whose 
exact value is yet to be determined, although the commonly 
close agreement indicates a most valuable conclusion, as 
will appear better in the Third Part. 

It is otherwise as regards high dilutions, at which, 
especially for strong monobasic acids, monacid bases, and 
their salts, the dissociation found osmotically and electrically 
is practically complete, and so ignorance of the law of 
dissociation does not stand in the way ; calculations then, 
with the assumption of complete dissociation, recover strict 

To give a motion of the case, we quote the degrees of 
dissociation found for decinormal solutions at 1 8 from 
conductivity measurements : 

Strong monobasic acids and monacid bases : 

HCl 94% KOH 93% NaOH 90% 

Salts of the above acids and bases : 

KCl 86% KN0 3 85% NaCl 84% NaN0 3 84% 

Salts of strong dibasic acids or diacid bases : 

Na 2 SO, 69% BaCl 2 7 5% 

Salts of dibasic acids and diacid bases : 

MgS0 4 45% CuSO, 39% 

Already in the copper salt we are dealing with a weak 
base ; if further a weak acid be chosen, as in copper acetate, 
the ionization is further reduced ; then, however, quite 
a different dissociation appears the hydrolytic which 
results in formation of free acetic acid and free base or 
basic salt, an effect that does not come into consideration 
with the salts mentioned above. 


3. SOLID SoLUTioxs 1 . 

There seems a prospect of extending the theory of 
solutions which allows of determining molecular weight in 
solution, to homogeneous solid mixtures. This possibility 
is conveyed by the expression ' solid solution ' for such 
homogeneous mixtures. 

According to the data which are at present available for 
testing this assumption, it seems necessary to distinguish 
between amorphous 'solid solutions/ such as e.g. the glasses 
which are homogeneous mixtures of silicates, and isomor- 
phous mixtures/ The latter are further removed from 
solutions just on account of their crystalline structure, 
whilst for the glasses there is a continuous series of states 
connecting the solid and liquid conditions that almost 
excludes any limit to the applicability of the laws of 
solution. Only the equilibrium reached by diffusion takes 
more and more time to reach on account of the increasing 
viscosity, and consequently the difficult problem in the 
8tudy of such solid solutions lies in the establishment of 
the necessary equilibrium. 

Qualitatively these solid solutions, both crystalline and 
amorphous, have much in common with liquid solutions, 
but on the quantitative side of the problem it is necessary 
first to keep to amorphous, or occasionally crystalline solid 
solutions, whilst isomorphous mixtures in the narrower 
senses will be discussed in the third division under molecular 

First, a few leading traits of a qualitative character may 
})e touched upon l . 

A. Qualitative Considerations. 

As regards qualitative features, first come the experi- 
ments on the lowering of vapour pressure of solids when 

1 Bodlander, Neues Jahrbuchf. Mineralogiej Geologiej Palaontologie, 1898. 

2 Van 't Hoff, Zeitschr. f. Phys. Chem. 5. 322. 


others form with them homogeneous or isomorphous 
mixtures. Von Hauer and Lehmann observed in this respect 
that the tendency of hydrated salts to effloresce, which 
indicates a somewhat high vapour pressure, is reduced by 
isomorphous mixture ; this was observed for lead sulphite 
in mixture with the corresponding calcium and strontium 
salts, iron alum with aluminium alum, copper formate with 
the formates of barium and strontium. The essential point 
is that each component shows a greater tendency to effloresce 
than the compound. 

This reduction in saturation pressure corresponds to 
the observation already mentioned (p. 61), that when the 
< Unsolved substance separates out along with the solvent 
as an isomorphous mixture., 
the depression is abnormally 
small. Thus e. g. in Fig. 1 1 
let AB and BC be the vapour 
pressures of the solid and 
liquid solvent respectively, so 
that B, their point of inter- 
section, represents the melting 
point T! . If now the solution, 
by taking up any non- volatile 
body, comes to have the smaller 
vapour pressure B 2 C 2 , the de- Fig. n. 

pression produced is T 2 T T ; 

if, however, the solid solvent also takes up some of the 
dissolved body, and so possesses the smaller vapour pressure 
A, BO, the depression is also less, viz. T 3 T 1? always on the 
assumption that the freezing point is the temperature at 
which solid and liquid have the same vapour pressure, 
which is certainly the case when the dissolved body is 
non- volatile. 

It may be remarked that just on this account the treat- 
ment of isomorphous mixtures is less simple. The close 
relation of the two bodies, as benzene and thiophene *, 

1 Van Bjlert, Zcitschr.f. Phys. Chem. 8. 343. 

To T. T, 


naphthol and naphthalene 1 , chlorobromo- and iodoform 2 ,&c., 
excludes at once the assumption that only one body, that 
present in great excess, the so-called solvent, is volatile. 
The condition of equilibrium at the freezing point is then 
no longer merely equality in the partial pressures of the 
solvent in the solid and liquid conditions ; only if the same 
equality of pressure holds for the dissolved substance do 
the conditions suffice for equilibrium. The question arises 
whether the fixed composition thus arrived at for the 
isomorphous mixture separating from a solution of given 
composition agrees with the facts, or, in other words, 
whether the cause that produces mixed crystallization does 
not involve a factor that influences the composition of the 
mixture. This is probable, on the one hand, because iso- 
morphous mixtures of all compositions cannot always be 
obtained (Part I, p. 55); on the other hand this may perhaps 
correspond to a simultaneous formation of two layers in 
the liquid. The whole thing awaits further investigation. 
Another point that may be added in considering the 
qualitative relations is that the diminution in vapour 
pressure which a solid suffers when a non-volatile body is 
taken up by it, involves a diminution of solubility, so that 
e. g. an isomorphous mixture should separate out when 
saturated solutions of alum and iron-alum are mixed. 

B. Quantitative Results. 

As regards the quantitative side of the problem the 
experimental data are not yet very complete or conclusive. 
The important common result that workers in this field 
have so far arrived at is that the solid state is not distin- 
guished by a more complex molecular structure, but that in 
solids, too, the dissolved molecule has as a rule the smallest 
size compatible with chemical facts, and at most double 
that size, as was found to be the case in liquid solutions. 
This result would be particularly important for isomorphous 

1 Kiister, Zeitschr.f. Pkys. Chem. 13. 452, 17. 355. 

2 Brunni, Atti R. Accad. del Linceij Roma, 7. 166. 


mixtures, because then it would be allowable to extend the 
conclusion from the small isomorphous admixture to the 
whole crystal, since in such a case all the crystalline 
molecules may well be assumed to have the same structure. 
On considering in more detail particular experiments, it 
will be well to treat separately the crystalline mixtures and 
the amorphous solutions. In the former we have to take 
account of the complication due to electrolytic dissociation, 
while in the latter we must bear in mind that a separation 
of mixed material, spoken of as a solid solution, may be the 
result of an alternate formation of layers, i. e. something 
quite different from solid solution. 

i. Isomorphoiis Mixtures of Electrolytes. 

In the region of solid solutions, as commonly elsewhere, 
the most complicated cases theoretically are the most 
accessible to measurement, and the first attempt to elaborate 
a method of molecular weight determination for solids 
referred to just this case of isomorphous mixtures. 

Nernst 1 made use of Roozeboom's * results on the com- 
position of the mixed crystals of thallium and potassium 
chlorate obtained on adding a known amount of potassium 
chlorate solution to saturated thallium chlorate. It was 
found that the concentration of the undissociated potassium 
chlorate in solution (normality c) was nearly proportional 
to that of the total potassium chlorate in the mixed crystal 
(x in molecular percentage). 




0.0168 2 8-4 

0-0873 12.61 6-9 

0-1536 25-01 6-1 

Applying the law of partition stated on p. 53, as regards 
the proportion of a substance in two non-miscible liquid 
layers, it may be concluded from this that potassium 
chlorate in the isomorphous mixture has the same mole- 
cular magnitude as in the solution, i. e. KC1O 3 . 

1 Zeitschr.f. Phys. Chem. 6. 577, 9. 137. 2 1. c. 8. 516. 


Fock 1 eliminated as far as possible the influence of 
electrolytic dissociation by using solutions of nearly the 
same total concentration obtained by adding a great excess 
of a salt containing the same ion ; so that if the elec- 
trolytes dealt with are equally strongly dissociated, the 
concentration of each dissociated part remains the same. 
In these cases the concentration in the solution (c l in nor- 
mality) usually appeared proportional to that in the mixed 
crystals (x in molecular per cent.). An experiment carried 
out at 25 with thallium nitrate may serve as example ; 
the solution contained the amount of potassium nitrate (c. 2 
in normality) necessary to keep up the total concentration. 

c i 

c i C 2 q + Ca XL - 

0-o8 O-IIT 

0-20 O-Il6 

0-57 0-116 

1-78 0-105 

2-19 0-109 

2-77 0-117 

Here, as in most other cases, the conclusion to be drawn 
is in favour of simple molecular magnitude in crystals. 

2. Crystalline Mixtures of Non-electrolytes. 

Here we come across, on the one hand, the most striking 
confirmation of the conception of solid solutions, especially 
as shown by van Bylert, Beckmann, and Brunni, and on 
the other a peculiar behaviour noted by Kiister. The 
observed data are not sufficient to allow of characterizing 
the two categories with certainty ; in the first case it 
seems as if any relation of crystalline forms, e. g. at least 
between iodine and solid benzene, were excluded, whilst in 
the latter case, e.g. between hexachlor- and pentachlorbrom- 
ketopentane (C 5 C1 6 O and C 5 Cl 5 BrO), far-reaching isomor- 
phism is conceivable ; we will, in the first place, therefore, 
distinguish ' crystalline solid solutions ' and ' isomorphous 

1 Berl. Ber. 28. 408, 2734 ; 31. 160, 506 ; Zeitschr.f. Kryst. 28. 337. 

o 0089 

















3-5 8 9 6 


Crystalline solid solutions. The two cases so far studied 
are of iodine in benzene and thiophene in benzene 1 . The 
benzene crystallizing out in each case is found to contain 
iodine or thiophene, and quantitative observations showed 
proportionality between the concentrations in the solid 
solution and liquid. 


Liquid .... 3-39 per cent. 2-587 per cent. 0-945 per cent. 

Solid solution . 1-279 0-925 0-317 

Katio .... 0-377 0-358 0-336 

Mean 0-357 

For thiophene the ratio only varied from 0-396 to 0-379 
with the maximum intervening value 0-449 ( mean OM- 1 ^ 
for solutions containing 1-16 to 15-91 per cent, of thiophene. 

From this proportionality it may be concluded, according 
to p. 72, that iodine and thiophene have the same molecular 
weights in solid and liquid benzene, i. e. the weights corre- 
sponding to the formula I 2 and C 4 H 4 S. 

At the same time we have a means of testing further 
quantitatively the considerations on the lowering of the 
freezing point given on p. 71. It appeared there that joint 
crystallization reduced the depression of the freezing point 
from T 2 Tj to T 3 T : (Fig. n). The quantitative relations 
are clear from the figure, viz. : 

TJ T 3 : T! T 2 = cB^ : cB 2 cd : c B = cB dB : cB, 

where cB and dB are the lowerings of vapour pressure 
suffered by the liquid and solid respectively. These, if the 
dissolved substance has the same molecular weight in each, 
are proportional to the concentrations. Hence 

/ dB\ 
TI T 3 = T x T 2 (i - -g) = 0-643 T i T 2 and 0-55 1 T! T 2 

respectively for iodine and thiophene. 

The molecular depressions will therefore be only 0-643 

1 Van Bylert, Zeitschr. f. Phys. Chem. 8. 343 ; Beckmann and Stock, 1. c. 
17. 1 20, 22. 609. See further Kiister, 1. c. 17. 364. Wiirffel, Beitrdge 
zm Molekulargewichtsbestimmung in krystallinischen Substanzen, Inaug.-Diss., 
Marburg, 1896. 


and 0-551 of the normal value 50 for iodine and thiophene. 
They were found : 

In 100 gm. (p) Depression (#) MoL lowering M 

Iodine (M = 254) . . 0-914 0-129 36 \ 

. . 2-24 0-313 35 | instead of 32 

,, . . 4-27 0-601 36 \ 

Thiophene (if =84) . 0-51 0-192 31' 

. 1. 12 0-422 31 

. 2-16 0-812 3r instead of 28 

9i - 3-25 1-213 

A similar confirmation is supplied by Brunni's 1 experi- 
ments on iodoform dissolved in bromoform. Here again 
bromoform crystallized out with the dissolved body, and 
the constant ratio (0-35 to 0-37) allowed of the conclusion 
that solid iodoform consists of HCI 3 , as in liquid solution. 
The molecular depression was found as 066 of the normal 
amount, while 0-65 to 0-63 was to be expected. 

Isomorphous mixtures. Against this result, in all 
respects satisfactory, has to be set the fact, observed 
especially by Kiister l , that isomorphous mixtures sometimes 
behave quite differently as regards melting point, which 
may be calculated from the simple rule of mixture 

8 = 0,8! + (i -a) S 

where $ x , S 2 are the melting points of the components, and 
a, i a the quantities (in molecules) of the two respectively. 
Thus for hexachlor- and pentachlorbrom-ketopentane the 
following results were found : 

Molecules Melting point Melting point 

C & Cl 6 0(a) C 5 a 6 rO(i-a) (o&s.) (caZc.) 

i o 87.5 (S x ) 

0-9471 0-0529 87.99 88-04 

0-9135 0-0865 88-3 88-38 

0.8571 0-1429 888 88-96 

0-8253 01747 89.11 89-28 

0-7468 0-2532 89-85 90-09 

1 Atti R. Accad. del Lincei (5) 7. 166. 

2 Zeitschr.f. Phys. Chem. 5. 60 1, 8. 577. 


Molecules Melting point Melting point 

C S CZ 6 (a) C.Cl.BrO (i - a) (oZw.) (caZc.) 

0-7005 02995 90-3 90-55 

0-5774 0-4226 91-61 91-81 

0-4109 0-5891 93-27 93-51 

0-2867 0-7133 94-59 94-78 

0-1791 0-8209 95-74 95-88 

0-0955 0-9045 96-67 96.74 

002OO 0-9800 97-49 97'5 

o i 97-71 (S a ) 

The same behaviour was found for some other pairs of 
very similar bodies. Still exceptions are found * which 
increase with the difference between the two melting points, 
and the abnormally large depression to be expected in the 
higher melting substance, when the difference of melting 
points is great, does not occur, e.g., with carbazol and 
phenanthrene. On the other hand, there is the character- 
istic rise of melting point, observed by Kiister, e.g. for 
/3-naphthol in naphthalene, accompanying the separation of 
a mixture richer in /3-naphthol 2 in accordance with the 
theory of solutions. It is so far not certain that Kiister 's 
pairs of substances form any departure from the theory. 

3. Amorphous Solid Solutions. 

Amorphous solid solutions would well form the first 
subjects of experiment in this department. Kiister 3 has 
made experiments on the partition of ether between water 
and caoutchouc, and he concluded with considerable proba- 
bility from the results, that ether in caoutchouc has the 
simple molecular weight when dilute, but double when 
more concentrated. 

The choice is, however, limited by the facility amorphous 
solids have of taking up substances in quite another way 
than mixture or solution. Thus the absorption of gases and 

1 Garelli, Gazz. Chim. ital. 1894, u. 263. 

2 Brunni, Atti E. Accad. dei Lincei, 1892 (2), p. 138. 

3 Zeitschr.f. Pkys. Chem. 13. 457. 


dissolved bodies by animal charcoal *, of dyes by fibres 2 , 
and so on are rather to be treated as surface phenomena. 

We will choose the simplest of such compounds for 
detailed examination the so-called palladium hydride 3 . 
What essentially concerns us in the research is the change 
in the amount of hydrogen absorbed by palladium at 
constant temperature, when the pressure, and therefore 
the concentration, of the hydrogen is changed, as well as 
the concentration of the solution of hydrogen in palladium 

Fig. 12. 

formed. The apparatus (Fig. 12) allows of bringing a 
known weight of palladium in e at definite temperature 
into contact with hydrogen whose pressure could be read 
off the manometer, whilst the quantity absorbed could be 
estimated from the original amount and the amount 

1 Schmidt, Zeitschr. f. Phys. Chem. 15. 56. 

2 Witt, Farbenzeitung, 1890-91, No. i ; Walker, Appleyard, Journ. Chem. 
Soc. 69. 1334. 

3 Hoitsema, Zeitschr. f. Phys. Chem. 17. i. See also Shields, Proc. Roy. Soc. 
Edinb. 1897-98, p. 169. 



remaining: the quantities being calculated by means of 
the known volume of the apparatus (provided with marks 
at k, i, k) and the pressure and temperature. 

It was already known, from the experiments of Troost 
and Hautefeuille, that the amount of hydrogen absorbed 
first rises with the pressure, whilst on further absorption 
the pressure becomes constant, which suggests the forma- 
tion of a compound. We may quote the data for 150; 
the pressure (P) in millimetres, the number of hydrogen 
atoms absorbed by one palladium (H), and the ratio between 
the increase of pressure (AP) and the quantity absorbed 

Pressure (P) Amount of absorption (H} 

AP : AH 

26-2 mm. 00097 


























0- 2 53 

















Troost and Hautefeuille's opinion in favour of the forma- 
tion of a compound was strengthened by the fact that 
at the temperature they employed the maximum hydrogen 
taken up in the interval of nearly constant pressure cor- 
responded to the formula Pd 2 H. It appears, however, from 
the newer researches discussed here, that this quantity does 
not represent any definite atomic ratio, and especially that 
it varies with the temperature. 

At io concentration about Pd H . 6 . 
180 PdH . 37 . 

The possible explanation of these facts is that at constant 
pressure two solid solutions are present in the palladium. 


which behave like two liquid layers l . Thus e. g. a similar 
behaviour as regards pressure would be found if unsatu- 
rated ether vapour were compressed in contact with water. 
At first the pressure will rise, with increasing absorption 
of ether by the water ; then a second layer will be formed, 
very rich in ether, i. e. moist ether, and then the pressure 
will remain constant whilst the new layer increases in 
bulk at the cost of the old. When the latter has vanished 
the pressure again rises. The analogy with two liquid 
layers has been actually found in solid solutions, in 
isomorphous mixtures as a matter of fact 2 . Beryllium 
sulphate and selenate, e.g., do not mix isomorphically in 
all proportions, but between S : Se = 7-33 : i and S : Se 
= 4:1 there is a gap. So that from a solution containing, 
say, S : Se = 5 : i two isomorphous mixtures will crystallize 
out, one tetragonal (7-33 : i), the other rhombic (4 : i). 

In following out further the behaviour of palladium 
hydride with regard to the molecular character of the 
hydrogen, only the parts of the series of observations 
before and after the region of constant pressure are 
available, and especially the former, because it refers to 
small concentrations of hydrogen, and so satisfies the con- 
dition implied in the theory of solution. 

According to what precedes (p. 28), if the pressure and 
the concentration of hydrogen in palladium are proportional, 
the formula H 2 is probable for the dissolved gas. The 
following results are obtained, with the aid of the density 
of palladium hydride :-- 



1 According to recent observations by v. Jiiptner on steel, there are 
indications of a similar phenomenon involving solutions of carbon in iron. 
' 2 See also Part I, p. 55. 

Pressure in 

C.c. Prf, containing 

mm. (p) 

2 mg. H (v) 















It appears that the concentration (--) is not proportional 

to the pressure, for - = pv (where v c.c. Pel, containing 
2 nig. H) varies from 808 to 303-5. On the other hand, 
v Vp is remarkably constant, as the fourth column shows. 
This means proportionality between the pressure and the 
square of the concentration, and from what precedes, makes 
it probable that the hydrogen is not at present dissolved 
as H 2 but as H, thus confirming the main conclusion that 
the solid state is not due to great molecular complexity. 


(Isomerism, Tautomerism) 

WHILST so far we have been concerned with the size of 
the molecules, we have now to consider their internal 
structure. The necessity for going into this point lies in 
the existence of isomerism. From the moment that bodies 
of the same molecular magnitude and composition with dif- 
ferent properties are known, such as ethyl- and dimethyl- 
amine, which, it is well known, both have the formula 
C 2 H 7 N, the possibility of different arrangements suggests 
itself as an explanation. For that reason the development 
of the theory of molecular structure has taken place in 
organic chemistry. Isomerism among inorganic compounds, 
such as between ammonium phosphite and hydroxylamine 
hypophosphite, both NH 6 PO 3 , is rare l . 

In the methods now to be described for arriving at the 
molecular structure, the molecular formula, which gives 
the number and nature of the atoms in the molecule, serves 
as a starting-point. It may be obtained, as mentioned 
on p. 14, from the qualitative and quantitative composition 
and the molecular weight. 

The further step towards molecular structure rendered 
necessary by isomerism is to find the mode of connexion 
of the atoms, a problem known as that of determining the 
structure or constitution. Then, the information derived 

1 Sabanejeff, Berl. Ber. 30. 285. See also the comprehensive investigations 
of Werner on metallic ammonium derivatives (Zeitschr. f. anorg. Chem.') ; 
also Kurnakow, Berl. Ber. 1898, p. 207. 


in this way proving insufficient, conceptions going in 
further detail into the relative positions of the atoms were 
developed as a study of configuration or stereochemistry. 
We shall here discuss each in turn, and in a third section 
the very peculiar phenomenon known as tautomerism. 


The study of constitution, then, refers to the mode of 
connexion between the atoms. In it the molecule is sup- 
posed motionless, so that the theory could at most only 
represent the facts accurately at the absolute zero. 

Of the methods in question which allow of deter- 
mining the mode of connexion, there are, of special 
importance : 

A. Determination of constitution on the basis of the 
valency of the elements combined. 

B. Determination of constitution from formation out of, 
and conversion into compounds of known structure. 

It may be added that many determinations of consti- 
tution are based on analogy, and therefore on the relations 
between physical and chemical properties and structure. 
But these will, according to the arrangement chosen, find 
their natural place in Part III. 

A. Determination of Constitution on the Basis of the 
Valency of the Elements combined. 

The notion of valency, on which this method is based, 
must first be quite briefly considered in its origin and extent. 
It is worth while then to glance at the binary hydrogen 
compounds of known molecular weight, in which only one 
atom of the other element occurs in the molecule, as : 

CH 4 , NH 3 , OH 2 , C1H, SiH 4 , PEL, SH 2J IH, &c. 

It appears then that never more than one atom of another 
element is combined with a single atom of hydrogen. This 
fact is expressed in the conception that hydrogen has only 

F 2, 


one faculty of combining, only one valency, as shown 
graphically by a stroke : 


On the assumption that this faculty of combination is in 
the formation of the molecule used, and used up, mutually, 
it follows that all elements which combine with hydrogen 
according to the symbol 

XH (C1H), that is X H (Cl H), 

like chlorine, bromine, iodine, &c., have, like hydrogen, only 
one valency, or are univalent. They can, therefore, like 
hydrogen, be used to determine the valency. The elements 
that combine according to the schemes 


are accordingly bi-, tri-, and quadrivalent. 

Simple as the fundamental conception of valency appears, 
its further development is complicated because there are 
elements which do not always appear with the same 
valency, e. g. iron is bi- or trivalent according as it occurs 
in the so-called ferrous or ferric compounds. Valency 
would accordingly supply a very unreliable basis for 
determination of constitution were it not that in the 
department in which such determinations are important, 
namely, in organic chemistry, the elements mainly con- 
cerned do possess constant valencies. Hydrogen appears to 
be invariably univalent, oxygen almost always bivalent, 
carbon quadrivalent, so the somewhat empirical method 
practically followed on these grounds is of the highest 
importance, and usually leads directly to the object aimed 
at. Thus a compound with the molecular formula CH 5 N 
can only, remembering the quadri-, tri-, and uni valence of 
carbon, nitrogen, and hydrogen, have the constitution 



In other cases, such as chloral, C 2 C1 3 OH, more than one 
possibility exists ; these may be conveniently arranged by 
separating the multivalent from the univalent elements, 
thus : 

(C 2 0)C1 3 H, 

and then calculating the number of bonds by which the 
skeleton of multivalent atoms is held together. Since each 
bond involves the use of two valencies, 

Total number of valencies in C 2 O .... 10 
Deduct for combination with C1 3 H ... 4 

Remainder 6 = 3 bonds. 

Hence the possibilities for C 2 O are 

(j) C = C O (2) C-C = O (3) C-C 


to which the chlorine and hydrogen atoms are to be added, 

= C O Cl (i 6)j}>C = C-0 Cl 
Cl H 

= C O H 


CK Cl\ 

(2 a)Cl-"C C = (2 6)C1SC C=O 


H/ I Cl/ | 

Cl H 


The choice between these six possible formulae must be 
made on other grounds. 

Simple as the method is, and valuable in cases such as 
the above, in which the valency of the elements combined 
is hardly open to doubt, it must be remarked that it is 
restricted in application, owing to uncertainty in the 


valency : it may often be replaced or supported by methods 
to be described later, which then show their scientific 
superiority in the elucidation they give to the notion of 

B. Determination of Constitution from Formation out of, 
and Conversion into Compounds of known Structure. 

In order to adhere to a definite example in this case 
too, we will discuss the constitution and isomerism of 
methyl mustard oil and methyl thiocyanate. The two 
compounds are known to have the same molecular formula, 
C 2 H 3 NS, and it was the relations of composition and 
decomposition that led Hof mann l to choose the appropriate 
constitutional formulae. 

Formation of methyl mustard oil takes place, amongst 
other ways, from methylamine (CH 2 NH 2 ) and carbon 
sulphochloride (C1 2 CS), with separation of hydrochloric 
acid : 

H 2 CNH 2 + C1 2 CS = aHCl + C 2 H 3 NS (methyl mustard oil); 

decomposition, e. g. under the influence of hydrogen 
(sodium amalgam and alcohol), to formation of methylamine 
and methylene sulphide (H 2 CS) : 

C 2 H 3 NS (methyl mustard oil) + 2 H 2 = H 3 CNH*+ H 2 CS. 


For the formation of methyl thiocyanate, on the other 
hand, we may use methyl sulphide (H 3 CSCH 3 ) and broin- 
cyanogen (CNBr). The thiocyanate is produced according 
to the equation 

H,CSCH 3 + BrCN = H 3 CBr + C 2 H 3 NS (methyl thiocyanate) ; 

while decomposition by means of hydrogen leads to separa- 
tion of methylmercaptan (H 3 CSH) and hydrocyanic acid 
(HCN) : 

C 2 H 3 NS (methyl thiocyanate) + H, = H 3 CSH + HCN. 

1 Beilstein, Handbuch der organ. Chemie, p. 699. 


All these reactions lead to a common characteristic of 
the constitution of the isomers, viz. the presence of a 
methyl group (CH 3 ), which is regularly contained both 
in the materials required for formation and the products 
of decomposition. The constitution of each is therefore 
definite to the extent of being 

(H 3 C)CNS. 

The difference is now simply shown in the above reactions, 
both by the materials required for formation and by the 
products of combustion, to lie in the fact that the methyl 
group of the mustard oil is connected to nitrogen ; but that 
of the thiocyanate to sulphur. This may be expressed by 
the constitutional formulae 

H ; ,CNCS(methylmustardoil) ) H 3 CSCN (methyl thiocyanate). 

The following schemes will then serve to express the 
reactions : 

For methyl mustard oil : 

H 3 CNH 2 + C1 2 |CS and 

For methyl thiocyanate : 

H 3 CS|CH3_4 1 Br 1 CN and 

It should be noticed that in this choice, made on the 
ground that the formation and decomposition can easily 
be expressed, something has been assumed as to the course 
of the reactions, viz. that in reacting, a molecule changes 
as little as may be. To illustrate this by an example, 
we may note that hydrocyanic acid might conceivably be 
formed from a body H 3 CNCS in this manner : 

H 3 C 

But then a double break would occur in the original 


molecule H,CNCS, whilst in formation from H 3 CSCN only 
a single one, as shown by the symbols 

H 3 C I NO I S and H 3 CS I ON. 

O I O 

Intramolecular atomic displacements. The question 
thus becomes very important, whether this assumption as 
to the mechanism of reaction always corresponds to the 
facts. With regard to this, a distinction must first be 
made between inorganic and organic, i. e. carbon compounds. 
In the former, which show the phenomenon of isomerism 
exceptionally, or not at all, the product is independent 
of the original condition of the components ; e.g. barium 
sulphate (BaSO 4 ) from oxide (BaO) and sulphur trioxide 
(SO. } ) is identical with that from the superoxide (Ba0 9 ) 
and sulphur dioxide (SO 2 ) ; here the peculiar rigidity which 
gives the compound the characteristic of its origin is 
absent. In organic compounds, however, with the appear- 
ance of isomerism, this is not the case. And that the 
rigidity is such that only the smallest possible change 
occurs in reactions, as in the case of mustard oil discussed 
above, is rendered probable by the fact that different 
determinations of constitution, all based on this principle, 
yield the same result : thus the formation and decomposition 
of methyl mustard oil lead to the same conclusion, and 
several other methods of formation and decomposition not 
mentioned here. There are cases, however, in which such 
agreement is wanting. 

We may first consider such a case, which was at first 
difficult to understand, but was afterwards fully explained, 
so affording a prospect of removing other uncertainties 
previously existing. The body in question is allyl thio- 
cyanate, NCS(C 3 H 5 ). The generally applicable method of 
preparation of thiocyanates, in which potassium thiocyanate 
(NCSK) is treated with the iodide of the radicle it is 
desired to introduce in this case allyl (C 3 H 5 ) leads to a 
body of the required composition, according to the equation 

NCSK + IC 3 H 5 = NCS (C 3 H 5 ) + KL 


It however appears to be allyl mustard oil, SCN(C 3 H 5 ), 
instead of the thiocyanate, NCS(C 3 H 5 ). The explanation 
was given that this body is the result of a secondary 
reaction due to the somewhat high temperature (100) 
employed. In the cold the allyl thiocyanate expected is 
actually formed, but it has the property of suffering an 
inversion on heating, i. e. being converted into the isomeric 
mustard oil. Two facts are thus simultaneously established: 
first, that an inversion of atoms within the molecule may 
stand in the way of determining constitutions by the 
method we are discussing; second, that to exclude such 
secondary changes, high temperatures should be avoided in 
determining constitution. 

It has not so far always been found possible to analyse 
these modified reactions, and distinguish with certainty the 
primary reaction to be arrived at ; e. g. pinacone, which has 
the constitutional formula 

(CH 3 ) 2 COHCOH(CH 3 ) 2 , 
is converted by withdrawal of water into pinacolin 

C 6 H 14 2 -H 2 = C C H 12 0, 
which accordingly was regarded by Friedel l as 
(CH 3 ) 2 C-C(CH 3 ) 2 



But as Butlerow obtained the same body from trimethyl- 
acetyl chloride, (CH 3 ) 3 CO Cl, and zinc methyl, Zn(CH 3 ) 2 , 

the formula 

(CH 3 ) 3 CCOCH 3 

was equally probable, according to the equation 

2(CH 3 )CCOCl-fZn(CH 3 )=:ZnCl 2 + 2(CH 3 )CCOCH 3 . 

In one of the two reactions an extra methyl group must 
have been detached and displaced, and so far it is an open 

1 See among others Pomeranz, Wien. AkacL Ber. 106. 579 (1897). 


question whether a subsequent process comes into play, or 
whether the displacement occurs during the reaction itself. 
We may conclude hence, for the practice of determining 
constitutions, that a formula arrived at only in one way is 
uncertain, but that as such inversions are rare, when two 
processes yield the same result with regard to constitution, 
the uncertainty is small. The conclusion is strengthened 
when one process is one of formation, the other of 
decomposition, since then the possibility of the same 
subsequent change influencing the reaction is excluded. 
In this way the above case of pinacolin was decided ; its 
decomposition on oxidation with formation of trimethyl- 
acetic acid, C(CH 3 ) 3 COOH, together with formic acid, 

C 6 H ]2 O + 3 O = C(CH 3 ) 3 COOH + HCOOH, 

has led to the adoption of the formula 
(CH 3 ) 3 CCOCH 3 . 

The result would be still more conclusive if the existence 
of an isomer rendered impossible the choice of a second 
formula, i. e. in the case considered, if the body 

(CH 3 ) 2 C-C(CH 3 ) 2 


were known and turned out to be different from pinacolin. 
Hence the case quoted on p. 86 of methyl mustard oil and 
methyl thiocyanate affords the greatest certainty, for 
formation and decomposition yield the same formula, and 
the other constitutional formula suits the mode of formation 
and decomposition of an isomeric compound. 


Whilst the study of constitution is restricted to 
determining the internal relations or bonds of the atoms in 
the molecule, the higher problem of finding the relative 


positions of the atoms, i. e. the structure of the molecule, 
in the literal sense of the word, is included under 

We will first describe the methods applied, and then 
devote a special section to stereochemistry, which developed 
through their application. 

The two methods are essentially the following : 

A. Determination of configuration by the number of 
isomeric derivatives. 

B. Determination of relative distances in the molecule. 
The first method especially has led to valuable results ; 

the second is at present chiefly of importance for its 
intrinsic scientific meaning. 

A. Determination of Configuration by the Number of 
Isomeric Derivatives. 

The wide applicability of the principle now to be 
discussed will be best appreciated by particular examples, 
of which cases may be chosen illustrating the fundamental 
meaning of the method. Such examples are those referring 
to the constitution of the benzene derivatives, and to 

i. Constitution of Benzene 1 . 

The molecular formula of benzene, C 6 H 6 , allows of 
a large number of structural formulae in accordance with 
the indications of the theory of valency, and possessing an 
equal degree of probability. Several isomers, such as 
dipropargyl, are in this way possible. 

The choice was made by Kekule on the ground of the 
number of isomeric derivatives, in particular because the 
monosubstituted bodies C 6 H 5 X, such as phenol. C 6 H 5 OH, 
exist in only one form, the disubstituted C G H 4 XY, such as 
oxybenzoic acid, C 6 H 4 (OH)(COOH), in three isomeric forms. 

1 Marckwald, Die Benzoltheoriej 1897 ; Vaubel, Der Benzolkern, 1898. 


To decide more closely, we may add the fact that 
substituted benzenes are not capable of decomposition into 
pairs of active isomers l . 

Kekule, in stating his view, relied empirically on the 
available experimental data. Ladenburg 2 gave later the 
strict proof, which will be repeated here. 

(a) Benzene gives only a single wwnosubstituted deriva- 
tive. Let us distinguish the six hydrogen atoms in the 
following way : 

C 6 H a H b H c H<fH 6 H/, 

and regard phenol (C G H 5 OH) as 

& H c H d H H. 

From phenol maybe derived abenzoic acid, C 6 H 5 COOH, 
by replacing the hydroxyl group with chlorine (by means 
of PC1-), the chlorine with methyl (by means of CH 3 I and 
Na), the methyl with carboxyl (by oxidation) ; this benzoic 
acid therefore has the COOH group in the a position. 
The three oxy benzoic acids, C ( .H 4 OHCOOH, derived from 
this acid, i. e. salicylic and meta- and para-oxybenzoic acids, 
differ on account of the OH group having replaced different 
atoms of hydrogen, and not that in a, therefore say those 
in by c, d, yielding : 

C 6 (COOH) a (OH) t H^ f , C 6 (COOH) ft H b (OH) c H,_ /5 
C fl (COOH) a H 6 , e (OH) d H ei/ , 

If now the calcium salts are distilled with lime, the 
carboxyl group is replaced by hydrogen, and we get the 
phenols : 

C 6 H a (OH) 6 H^ C c H a . 6 (OH) c H,,_ /F C e H a , & , (OH) ll H 1 . 1/ , 

which Ladenburg found to be identical with the original 
C C (OH),,H 6 _, 

Thus substitution of the hydrogen atoms a to d in 

1 Van 't Hoff, Atornlagerung im Eaume (2nd ed.), p. 92. 

2 Theorie der aromatischen Verbindutigen, 1876. 


benzene leads to the same product ; this may be expressed 
symbolically as 

H a = H,= H c = H, (i) 

In the second place it appears that in benzoic acid there 
are still two hydrogen atoms whose substitution leads to 
the same result. 

Let us take e. g. the oxyacid, C G (COOH) a (OH) 6 H c _ 7 , and 
convert the corresponding bromtoluene, C (CH 3 ) a Br 6 H c _ /? 
into nitrobromtoluene, C G (CH 3 ) a Br 6 (NO 2 ) H 3 ; this may be 
converted by reduction into an amidotoluene, C 6 (CH 3 ) a H b 
(NH 2 ) H 3 , and the latter by diazotizing and treatment with 
bromine, into C 6 (CH 3 ) a H 6 Br H 3 , which is identical with the 
original substance. Accordingly in benzoic acid, C 6 (COOH) a 
H 6 _/, one hydrogen atom is in a position identical with 6, 
consequently either 

H & = H c , H & = H d , H 6 = H e , or H 6 = H/. 

The first two possibilities, however, disappear, since the 
corresponding oxy acids that have the OH in 6, c, and d 
respectively are different, so there remain only the 

H b = H e and H 6 = H, .... (2) 

In the third place there is, besides H & , still a pair of 
similar H atoms in benzoic acid, as appears from the fact 
that the oxybenzoic acid in question, C 6 (COOH) a (OH) & H c _ /5 
gives a bromobenzoic acid, C ti (COOH) ft Br 6 H c _/, which leads 
to two isomeric nitrobromobenzoic acids, C (COOH) a Br 6 
(NO 2 ) H 3 , whose reduction gives the same amidobenzoic 
acid, C, (COOH) a H 6 (NO 2 ) H 3 . Thus (as again H c = H d is 
excluded on account of the difference between the corre- 
sponding oxybenzoic acids) either 

H c = H e , H c = H /? H, = H e , H, = H/, or R e = H 7 (3) 

If (2) and (3) are combined, remembering that equivalence 
of b, c, and d is excluded, so that e.g. the combination 

H, = H, and H c = H e 


in benzole acid is not allowable, the remaining possibilities 

are: H, = H e and H e = H, (4) 

H 6 =H e H^H, . . . . . (5) 

H t = H e H e = Hy (6) 

H t = H r H c = H e (7) 

H 4 = H, Hu = H. (8) 

H & = H, H. = Hr (9) 

These hydrogen atoms must, then, have identical positions 
in benzene, and we have already 

H ft = H 6 = H c = H, (i) 

so that combination of (i) with the possibilities given by 
(4) to (9) leads to the conclusion 

H a = H 6 = H c = H d = H e = Hy. 

(6) The second fact, the existence of three disubstituted 
products, also assumed by Kekule' on the ground of the 
data then existing, is included in the foregoing as a necessary 
consequence. Besides the three oxybenzoic acids, 

C 6 (COOH) a (OH) t H c _ /; C 6 (COOH) a H 6 (OH) e H,,., 
and C 6 (COOH) H,, c (OH) d H e>/ , 

two other isomers are conceivable, with the hydroxyl group 
in the place of the hydrogen atoms H g , H/ respectively. 
But we have seen that one of the six combinations (4) to 
(9) necessarily holds, and this makes the other two isomers 
impossible, since one of these equivalences must hold for 
benzoic acid, C 6 (COOH) a H 6 _/, viz. : 

H 6 = R e and H c = H, (4) 

HI, = H e H^ = H/ (5) 

H t = H. = Hr (6,9) 

H,, = H/ and H. = H e (7) 

H 6 = H/ H d = H e (8) 

(c) The third fact, that the benzene substitution products, 
not containing asymmetrical substituting groups, cannot be 
separated into optically active isomers (proved at least for 
the trisubstituted compounds), shows that compounds of the 





type C G (a b c d ef), i.e. with five hydrogen atoms replaced by 

different groups, C, HX 1 X 2 X 3 X 4 X 5 , 

have a symmetrical constitution ; for asymmetry implies 

a non-identical reflected image, so that we should expect 

two isomers differing as we shall see later by their 

opposite rotation of polarized light, their so-called optical 


This condition is only satisfied if benzene has a plane of 
symmetry, in which all the hydrogen atoms lie ; in other 
words, all the hydrogen atoms are in one plane, with 
respect to which the carbon atoms are symmetrically 
placed. The suggested spatial arrangements of the 
hydrogen atoms are thus 
excluded. With Laden- 
burg's prism formula, e.g., 
even the diderivatives 
C 6 H 4 X 2 would suffer de- 
composition into optical 
antipodes, since the two 
symbols in Fig. 13 are not 
symmetrical, and there- 
fore not identical reflected 
images ; the assumption 
of the regular octahedron with the hydrogen atoms at the 
corner points carries one a little further, but there again 
the triderivatives would suffer decomposition. 

How the hydrogen atoms are arranged in the plane 
follows partly from the first principle, that there is only 
a single monoderivative, according to which the hydrogen 
atoms must be similarly situated. Two possibilities occur : 














^ x 



m HC 



Fig. 13 




The second is the arrangement at the angles of a regular 
hexagon ; the first at those of a hexagon with alternately 


larger (2.3, 4.5, 6.1) and smaller (1.2, 3.4, 5.6) edges, 
very suggestive of Kekule's formula with alternate single 
and double linkages. 

The second principle, that three diclerivatives, C G H 4 X 2 , 
exist, settles the choice, since only the second possibility, 
the regular hexagon, provides the three isomers : 


H H H X and H H 

H H H H H X 

distinguished as i . 2 (ortho), 1 . 3 (meta), and r . 4 (para), 
whilst 1.5= 1.3 and i . 6 = 1.2. With the irregular 
hexagon we should have four isomers, since 1.5 = 1.3, but 
i . 6 is not = 1.2. 

The position of the carbon atoms is indeterminate, except 

that it must leave untouched the symmetry of the whole 

with respect to the plane in which the hydrogen atoms lie, 

and the identical position of the hydrogen atoms, whilst 

the number of diderivates remains three. An arrangement 

jj jj indicated by the formula for benzene 

C C satisfies these conditions completely, as 

H C C H does any in which the carbon atoms are 

C C in a regular hexagon in the same plane 

as the hydrogen atoms, and with the 

same centre ; either, therefore, the above, or the same figure 

rotated 30. The symbol may, without losing its validity, 

be brought into accordance with the theory of valency in 

either of the following ways : 

H H H H 

\ / \ / 

C C C C 

H C^X^C-II and H ( ^C-H 

\7\> V V 

C C C C 

/ \ / \ 

H H H H 

due respectively to Glaus and to Armstrong and Baeyer 
(the so-called centric formula) ; in the latter it is assumed 


that the fourth valencies of the carbon atoms do not 
saturate each other singly, but neutralize in their 

2. Determination of Position in the Benzene Derivatives. 

After the constitution of benzene was so far settled as to 
give an explanation of the existence of only one mono- 
derivative, but of three isomeric diderivatives, the question 
arose, which of the three possible constitutions i . 2 (ortho), 
] . 3 (meta), i . 4 (para) to attribute to any given dideriva- 
tive. This is the problem known as absolute deter- 
mination of position. Kekule considered the answer as 
only partly possible, regarding experiment as limited to 
determining whether two compounds belong to the same 
series or not. E. g. which dichlorbenzene is to be grouped 
with any one of the three dioxybenzenes say hydro- 
quinone can be found from the conversion of hydroquinone 
into dichlorbenzene by means of PC1 5 , 

taking care as far as possible to exclude an intramolecular 
displacement (p. 88). This is accordingly a relative deter- 
mination of position. 

The further problem of absolute determination is now to 
settle whether these two compounds, or rather the whole 
group they belong to, has the constitution i . 2 or another. 

Korner answered this question also by the method of 
the number of isomeric derivatives. He remarked that 
a diderivative, such as C G H 4 Cl 2 (i .2), leads on introduc- 
tion of a third group, say NO 2 , to two possible formulae, 
1.2.3 an< ^ i 2 . 4, as is shown by the diagrams : 

whilst 1.2.5 = 1.2.4 and 1.2.6 = 1.2.3. But C 6 H 4 C1 2 



(1.3) gives three isomeric triderivatives, viz. I . 3 . 2, i . 3 . 4, 
and 1.3.5. Finally, C 6 H 4 C1 2 (1.4) allows of only a 
single possibility, since 1.4.2 1 .4.3 = 1.4.5 = 1.4.6. 
The application of these principles to a very complete 
collection of experimental material resulted in an un- 
exceptionable solution of the problem, and one that accords 
satisfactorily with the indication of constitution given by 
the formation of benzene compounds out of, and their 
conversion into, fatty bodies. 

B. Determination of Relative Distances in the Molecule. 

The different atoms and groups in the same molecule 
may act on one another in a way that gives information as 
to their relative distances. This action may appear in two 
ways. First, it may lead to a reaction, caused by combina- 
tion and partial separation of the two groups ; but on the 
other hand the character of a group may suffer a change, 
which can often be followed out quantitatively. 

i. Mutual Action of Different Groups. 

An action that often occurs in the domain of organic 
chemistry is the combination of two hydroxyl groups with 
separation of water : 

X(OH) 2 = XO + H 2 0. 

Of this an example may be quoted. 

Many facts point to the part played by mutual distance 
in such actions, and especially, first, that in organic 
compounds, when two hydroxyl groups are attached to the 
same carbon atom, they at once suffer the change, or rather 
the existence of bodies containing the group C(OH) 2 is 
exceptional. Chloral hydrate, which appears to have the 
formula CC1 3 CH(OH) 2 , seems to be one of the few such 
compounds, whilst e. g. the attempt to form the correspond- 
ing compound, CH 3 CH (OH) 2 , leads to formation of aldehyde, 



CH 3 CHO, with separation of water. On the other hand, 
compounds containing the group C(OH)C(OH), such as 
ethylene glycol, CH 2 (OH)CH 2 (OH), are mostly very stable. 
It must be remarked, however, and it adds to the difficulty 
of making rational use of the principle, that not only 
mutual distance, but other influences as well, affect the 
separation of water from two hydroxyl groups, as is clear 
from the case of chloral hydrate 1 . 

An interesting example from the aromatic compounds 
may be added, in which the method discussed led to 
a direct and reliable determination of constitution. The 
three phthalic acids, C 6 H 4 (COOH) 2 , have the configurations 









HC 4 













Now only one of them easily forms the anhydride, 


C c H 4r ,~O, with separation of water ; this one was therefore, 

and correctly, assumed to be orthophthalic acid (1.2), whilst 
the other formulae were assigned to isophthalic (1.3) and 
terephthalic acid (1.4); in the latter the carboxyl groups, 
COOH, containing the hydroxyl, are more widely separated. 
It may be concluded in general that such condensation, 
i. e. formation of an internal anhydride with separation of 
water, occurs most easily in the orthoderivatives (1.2). 

1 See amongst others, measurements of velocity in the conversion of 
chlorhydrins. Evans, Zeitschr. f. Phys. Chem. 7. 356. 

G % 


2. Mutual Influence of Different Groups. 

The mutual action of two groups in a molecule may 
thus show itself in bringing about a combination, and 
chemical reaction between them ; but without going so far 
as that it may suffice to change the character of one of the 
groups under the influence of the other. We may take as 
instance the acid character due to the group COOH ; this 
property may be strengthened or weakened, for the reason 
considered. Since Ostwald's dissociation constant offers 
a ready measure of the acid character, the principle of 
determination of constitution has been tested in this 
direction by the author l named. The dissociation constant 
K which governs equilibrium between undissociated acid 
and ions, according to the equation 

and therefore bears the following relation to the concentra- 
tion of the undissociated acid C a and ions C- (Part I, p. 118), 

n 2 
IT - * 


may easily be determined from the conductivity, since the 
dissociated part is the same fraction of the total concentra- 
tion that the molecular conductivity /x for the concentration 
in question is of its limiting value /x w , i. e. 

Hence C i and C a are calculable for any given concentration, 
and therefore K is so : 

K -*- = 

whence a expresses the fraction dissociated into ions. 

1 Zeitschr.f. Phys. Chem. 3. 418. 



The values of K thus measured are suitable for judging 
of the influence we are considering, not only on account of 
the facility of determination, but because they vary so 
greatly amongst themselves. Thus the influence of chlorine 
at 25 in strengthening an acid is shown by the numbers : 

Acetic acid, CH 3 COOH iooK= 0-0018 

Monochloracetic acid, CH 2 C1COOH ... = 0-152 

Dichloracetic acid, CHC1 2 COOH .... 5-14 

Trichloracetic acid, CC1 3 COOH .... =121 

and in crotonic and isocrotonic acids, isomers of the formula 
CH 3 CHCHCOOH, chlorine shows, as might be expected, 
a greater influence on K when near than when far from the 
carboxyl ; the numbers again refer to 25 : 

a Chlorderivative (CH 3 CHCC1COOH 

Crotonic acid 




Isocrotonic acid 
o 0036 

Since we have here an indication of the possibility of 
determining relative distances within the molecule, we 
must mention the qualifications so far existing. 

On the theoretical side it must be remembered that, in 
consequence of the unequal distribution of matter in the 
molecule, an action is rarely propagated equally in all 
directions within it. 

On the experimental side it appears that sometimes 
a group in the meta-, sometimes one in the para-position 
behaves as if it were at the smaller distance. This is 
shown in the following ortho- (1.2), meta- (1.3), and para- 
(1.4) derivatives in the benzene series 1 . 

Chlor derivatives 



Benzoic acid 

Benzoic acid 



Benzoic acid 



iooA'= 0-006 
- 0-0093 

o 616 

o 0589 

0-0 4 I2 

o-o 7 5 

'ii I 
o-o 9 4 



1 Loewenherz, Zeitschr.f. Phys. Chem. 25. 385. 


The substituting group always exercises most influence 
in the ortho-position (1.2); benzoic acid is strengthened 
by chlorine and the nitro group to the greatest extent, 
and the latter too brings out the acid character of phenol 
and suppresses the basic character of aniline most in the 
ortho-position. But comparison of the meta- and para- 
derivatives leads to varying results, since chlorine exercises 
the greatest influence in the meta-, the nitro group in the 
para-position. The oxyderivatives appear to behave like 
those of chlorine, except that here the acid character is 
weakened in the para-position, while it is strengthened in 
the ortho- and meta-. 

Velocity of reaction, which often in other ways goes with 
this dissociation constant, but is not usually so easy to 
measure, shows the same peculiarities, so the velocity 
constant k for some reactions 1 may be given here (see 
Part I, p. 191). 

Combination of Bromallyl with 
Methylaniline (Toluidine) Chloraniline 

(1.2) .... fc= 54 9 

(1.3) .. . . fc = 4 45 23 

(1.4) . . . . k= 96 34 

( Aniline k = 68) 

Brommeihyl with 


(Aniline Jc = 24) 

The case is again simplest for the chlorderivatives : 
the velocity, 68 for aniline, is reduced, the most by chlorine 
in the ortho-position to 9, then in the meta- to 23, then in 
the para- to 34. 

C. Stereochemistry. 

Since constitutional formulae, expressing only the mode 
of combination of the atoms in the molecule, no longer 
sufficed to explain the cases of isomerism observed, a new 
attempt at explanation had to be made. Such a case arose 
in the discovery by Pasteur in 1853 ^ laevo-tartaric acid, 

but in the then state of the study of constitution was not 

1 Menschutkin, BerL Ber. 30. 2966, 31. 1423. 


recognized as such. It was Wislicenus who first, in 1873, 
moved by the discovery of the isomeric active ethylidene- 
lactic acids, CH 3 CHOHCOOH, expressed himself convinced 
that the usual constitutional formulae do not suffice to 
elucidate this isomerism. 

Some extension of the theory of constitution was thus 
necessary, and of the three possibilities, dissimilarity in the 
valencies of carbon, displacement of the atoms, and the 
relative position of the atoms in space, the last, as is well 
known, turned out to be the satisfactory explanation 1 . 

Dissimilarity between the different valencies of the 
carbon atom is not suited to supply the explanation, 
because that assumption would require isomerism in even 
the simplest derivatives of the type C(X) 3 Y, such as 
chlormethane, CH 3 C1. It is of importance that Henry 2 
carried out a systematic research on this point, preparing 
nitromethane, H 3 CNO.,, in four different ways, each of which 
would have brought the nitro group into the place of 
a different carbon atom. If, then, we distinguish the four 
hydrogen atoms by the indices abed, he prepared 

C(N0 2 ) fl H 6 c d , CH tt (N0 2 ) 6 H c * CH a 6 (NO 2 ) C H rf , 
and CH a6c (N0 2 ),, 

which turned out to be identical. 

The starting material for this was methyl iodide, 
CI a H 6 c d : this, combined directly with Ag NO 2 , was trans- 
formed into C(NO 2 ) a H &C rf. Another portion was converted 
by KCN into the nitril of acetic acid, C(CN) a H 6 c d , and then 
into acetic acid itself, C(COOH) a H fe c d ] this was chlorinated 
to C(COOH) a Cl 6 H crf , and on treatment with AgNO 2 
gave, through the transiently formed nitroacetic acid, 
C(COOH) a (N0 2 ) 6 H cd , the second nitromethane, CH a (NO 2 ) 6 
H cd . Another portion of the nitroacetic acid was then 
converted into malonic acid, chlorinated to C(COOH) a6 
Cl c H d , and reduced by AgN0 2 to the third nitromethane, 

1 Van 't Hoff, Lagerung der Atome im Raume (2nd ed.), 1894. 

2 Zeitschr.f. Phys. Chem. 2. 553. 


CH a6 (N0 2 ) c H ci . Renewed introduction of the carboxy] 
group, chlorination, and nitration yielded the fourth nitro- 
methane, CH abo (N0 2 )^. The four nitromethanes thus 
arrived at were found, as already remarked, to be identical. 

Further, we may briefly discuss the possible explanation 
of isomerism not falling within the theory of constitution, 
by differences in atomic movements. Berthelot l suggested 
it for the case of tartaric and lactic acids. The attempt, 
however, remained without success, as it was not possible 
to formulate the view clearly. It should be observed in 
this connexion that any attempt at explanation based on 
atomic movements requires that the phenomenon explained 
should vanish at the absolute zero, since there atomic and 
molecular movements cease. Fall of temperature should 
therefore lead to a gradual assimilation of the differences 
between isomers, if difference of atomic movement is the 
cause, and of this the isomers in question do not show the 
smallest indication. 

The third alternative, ilsomerism due to different positions 
of the atoms in space, has - shown itself capable of meeting 
the case, and led to the development of stereochemistry. 
We will bring the subject forward in the following 
divisions : 

(1) The asymmetric carbon atom and separation into 
optical antipodes. 

(2) Single carbon linkages, and compounds with more 
than one asymmetric carbon atom. 

(3) Double linkages and rings. 

(4) Stereochemistry of other elements. 

i. The Asymmetric Carbon Atom and Separation into 
Optical Antipodes. 

Stereochemistry is based on the principle (p. 91) of 
determining configuration by the number of isomeric sub- 
stitution products, supported and extended by application 

1 Bull, de la Soc. Chim. 1875. 

2 Walden, Journ. russ. Phys.-Chem. Ges. 30. 483. 


of the method (p. 98) depending on relative distances in the 
molecule. The fact is that in methane derivatives of the type 

in which, therefore, the carbon is combined with four dif- 

ferent atoms or groups, such as lactic acid, just mentioned, 


two (oppositely optically active) isomers regularly occur. 
The difference of the groups or atoms is essential, since, 
e. g., if in the above the hydroxyl group be replaced by 
hydrogen, both isomeric lactic acids lead to the same 
(optically inactive) propionic acid, 

C(CH 3 )H 2 COOH. 

A configuration of methane, such as is shown in Figs. 14 
and 15. with the four groups R in one plane and uniformly 
arranged, would not agree with the facts, for then the type 

with only two different groups should lead to isomerism. as 
shown by the difference between Figs. 14 and 15. But the 

R, - C R 2 Ra C - R 2 


Fig. 14. Fig 15. 

arrangement at the angles of a regular tetrahedron excludes 
isomerism in this and other cases, unless all four groups are 

Fig. 16. Fig. 17. 

different, as is indicated by the difference between Figs. 16 
and 17 ; this difference would obviously vanish if only R. 



and R 4 are alike. Not only the occurrence and non- 
occurrence of isomerism agree with this fundamental 
assumption, but also the character of the isomerism. In 
ordinary isomerism, as that of C 6 H 4 C1 2 (1.2) and (i . 3), there 
is a difference in every respect, in melting point, boiling 
point, solubility, density, and also in chemical behaviour. 
The two isomeric lactic acids, on the other hand, agree in 
almost all points, as in those just mentioned, in accordance 
with the complete equality of their internal dimensions. 
Differences occur only in respect to qualities that refer 
to the difference between two unsymmetrical reflected 
images, such as Figs. 16 and 17 are. 

The most noticeable difference is in optical activity in 
the dissolved or the liquid condition ; this shows itself in 
an opposite (and equal) rotation of polarized light, and 
every substance containing a carbon atom united to four 
different atoms or groups a so-called asymmetric carbon 
atom occurs in these two oppositely active forms, the 
optical antipodes. The simplest body yet known that 
illustrates this is chlorobromofluoracetic acid l , 

which has been obtained in two antipodes, and from which, 
presumably, the isomeric chlorobromofluormethanes, 


should be got by splitting off carbon dioxide. 

A second difference shown by the isomers in question is 
in their crystalline form. This 
expresses itself in the formation 
of two so-called enantiomorphic 
forms due to hemihedry such as 
Figs. 1 8 and 19 show for laevo- 
and dextro - ammonium bimalate. 
Fig. 18. Fig. 19. The two forms, like the diagrams 
Figs. 1 6 and 17, representing the 

atomic arrangement, constitute a pair of unsymmetrical 

1 Swartz, Bull, de I'Acad. de Bdg. (3) 31. 28. 


reflected images, that cannot be superposed on one 

The similarity of the antipodes in other respects implies 
the necessity of a separation, when a body with an asym- 
metric carbon atom is prepared in the laboratory. For 
starting with an inactive, symmetrical body, such as 

when a fourth different group R 4 is introduced in place of 
one R 3 , the two R 3 groups, being symmetrically contained 
in the molecule, are replaced with equal facility, and so an 
inactive mixture of the two isomers, CR 1 R 2 R 3 R 4 , in equal 
amounts is produced, in which the components are yet to 
be separated. It is well known that in the organism, on 
the other hand, the antipodes occur separately, e. g. tartaric 
acid obtained from grapes is the active dextro-rotary 

The methods of separation available may be grouped 
as follows : 

(a) Methods based on the phenomena of solution. 
(a) So-called spontaneous separation. 
(/3) Separation by means of active compounds. 

(6) Methods based on chemical behaviour, depending on 
the action of enzymes and organisms. 

(a) Methods based on the Phenomena of Solution. 

(a) Spontaneous separation. Separation of the optical 
antipodes is rendered difficult by their similarity of behaviour 
in almost every respect, especially their equal solubility; 
moreover, the isorners mostly unite to form a so-called 
racemic body, as in the classical example of the racemic 
acid precipitated on mixing concentrated solutions of the 
two oppositely active tartaric acids. 

The fact discovered by Pasteur, that the substances do 
not always appear together in this way, but that, e. g., from 
the solution of the sodium-ammonium salt of racemic acid 
laevo- and dextro-sodium-ammonium tartrates crystallize 


side by side, led to the first actual separation. This fact 
has since acquired a certain general interest in natural 
philosophy. Pasteur, who held the view that optically 
active compounds could only be obtained by the action 
of living organisms, attributed the so-called spontaneous 
separation to the action of germs in the atmosphere. 
Wyrouboff opposed this view, and since then the mechanical 
explanation of the separation has become clear, viz. that 
separation takes place regularly when the mixture of the 
optical antipodes is less soluble than the racemic substance. 
The phenomenon depends on temperature, and for some 
bodies occurs only under restricted conditions of temperature, 
as will be seen from the following more detailed account 
of particular cases. 

Stadel found that the solution, which in Pasteur's hands 
yielded the two sodium-ammonium tartrates, gave him only 
a sodium-ammonium racemate ; this was more thoroughly 
investigated by Scacchi, whilst WyroubofF found that the 
appearance of one or the other from solution depended on 
temperature ; that if supersaturation was avoided the 
mixed tartrates appeared below 28, the racemate above. 
Van 't Hoff and van Deventer 1 then showed that we have 
here to do with a transition (Part I, p. 25). Just as with 
Glauber salt, the separation of hydrate (Na 2 SO 4 . ioH 2 0) 
or anhydride, according as one works below or above 33, is 
associated with the complete transition of the salt at 33, 
according" to the equation ,// -. 

Na 2 S0 4 . ioH 2 = Na 2 S0 4 + ioH 2 O, 

so the alternative crystallization of mixed '* tartrates or of 
racemate is associated with a transition taking place at 28, 
according to the equation 

CC 4 H 4 6 NaNH 4 . 4 H 2 + t)C 4 H 4 O c NaNH 4 V4 H 2 O - 
(C 4 H 4 O fl NaNH 4 ) 2 2 H 2 O + 6 H 2 O; t \ 

The two phenomena may be observed in the/same manner. 

1 Zeitschr. f. Phijs. Chem. i. 173 ; van 't Hoff, Goldschm'idt,'Jorissen, 1. e. 
17-49- - ' " - 


The transition of Glauber salt appears as a melting point, 
and can be determined either as the temperature at which 
that salt melts, or that at which a mixture of anhydrous 
sodium sulphate and water freezes. Beckmann's apparatus 
for exact determination of melting points is admirably 
adapted to the observation. 

The same can be done for the formation and decomposi- 
tion of racemates, and in that way, e. g., the temperature 
(40) at which rubidium racemate splits up into the optical 
antipodes has been determined 1 . 

Since, however, a somewhat large amount of material 
is needed to determine transition temperatures in this way, 
the dilatometric ".process described in Part I, p. 25, in which 
the change of volume accompanying transition is used 
as indication, is often to be preferred. A dilatometer filled 
with the tartrate mixture, e. g., gave, as indication of the 
transition of the sodium-ammonium tartrates into racemate, 
the following readings of the level of the liquid : 

16 7 350 mm. l6 per ^ 

26 7 5io > 

27-7 3 6 72 - * 

3t-7 727 I H I D 

The conversion, therefore, took place between 26-7 and 
27-7, with an expansion equivalent to a rise of the liquid 
by 162 15=147 mm. 

As in the case of Glauber salt, so in that of the racemate, 
the transition is connected with a singularity in the solu- 
bility curve. This for Glauber salt shows a break at 33, 
due to the intersection at that temperature of two solu- 
bility curves, one referring to the hydrate, the other to the 
anhydride (Part I, p. 66). The following numbers, taken 
from Lowel, will serve to illustrate this ; they give the 
weight of anhydrous sodium sulphate per 100 parts of 
water in the saturated solution : 

Saturation for 31-84 32-65 34 

N^SOi.ioH.O 40 U9-78) 55 

J,2. ..: Na 2 S0 4 49.91 U9-78) 49-56 

1 Van 't Hoff, Muller, Berl. Ber. 31. 2206. 




The numbers show that below 32-65 the solution satur- 
ated for anhydride is supersaturated with respect to 
NaSO 4 . 10 H 2 O ; here we are clearly dealing with the 
well-known supersaturated Glauber salt solution. Above 
32-65 the state of affairs is reversed, and the solution 
saturated for the hydrate is the more concentrated. The 
two facts accord with the reversible transition in one or 
the other direction. 

The tartrate mixture and racemate show precisely similar 
behaviour. For 100 molecules of H 2 O there were found, 
expressed in molecules of C 4 H 4 6 NaNH 4 : 

For Saturation with 
Tartrate mixture 
Racemate . . . 



Only the solubility curve could not be followed out higher, 










, " 



7f ' 

^ - 














4 28 32 36 40 44 48 52 56 T 

Fig. 20. 

because a second transition into sodium and ammonium 
racemates takes place. For this reason rubidium racemate is 
better suited to show the point, having a corresponding tran- 
sition at 40. Measurements of solubility gave, expressed 
in molecules of C 4 H 4 O 6 Eb 2 for 100 molecules of water: 


Saturation with 25 35 40-4 40-7 54 

Tartrate mixture . 13-03 13-46 13 83 (Line A'B Fig. 20) 

Racemate . . . 10-91 12-63 13-48 . (Line AB ,, 

Whilst, therefore, comparison with a simple transition 
between two hydrates (as of Glauber salt) brings out 
evident relationships, there is a difference to be noted, 
for the formation or decomposition of a racemate is not 
a simple conversion of one salt into another, but of a salt 
mixture (tartrates) into a single salt (racemate) or vice versa. 
Remembering this, it is clearly the formation or decomposi- 
tion of double salts that should be quoted as analogy, and 
we will therefore describe what is essential in the relations 
of solubility at the transition temperature in such cases. 
(See Part I. p. 81.) Let us take the formation of astrakan- 
ite, the sodium-magnesium sulphate, Na 2 Mg(SO 4 ) 2 . 4 H 2 O, 
from its components, Na 2 SO 4 . 10 H 2 O and MgSO 4 . 7 H 2 O, 
which takes .place at 22 according to the equation 

Na 2 S0 4 . ioH 2 O + MgSO 4 .7H 2 O = Na 2 Mg(S0 4 ) 2 .4H 2 O 

Here, again, we have an apparent fusion that may be 
followed by the thermometer or dilatometer. 

We now find, in the behaviour with regard to solution, 
new relations to the formation of a racemate. These 
appear in the fact that at 22 three solubility curves inter- 
sect. From low temperatures up, we have the solution 
saturated for the two sulphates (i); at 22 formation of 
astrakanite sets in, and according as there is excess of the 
sodium or magnesium salt, we reach either the curve of 
saturation for astrakanite and sodium sulphate (2), or that 
of astrakanite and magnesium sulphate. Schematically : 

Na 2 S0 4 . ioH 2 , 2 ) Astr . + Na go . 1O H 2 O 

+ MgS0 4 . 7 H.O (i) ( 

22 \(3) Astr. + MgS0 4 . 7 H 2 O 

The same is to be expected for the formation of racemate. 
Thus, for the sodium-ammonium salt from low temperature 


to 27 the saturation curve for the tartrate mixture 
(i) holds; at 27 formation of racemate occurs, and if the 
two tartrates are present in equal amounts, we obtain 
the former racemate curve. If one or other tartrate is 
present in excess, we come in the one case to the curve of 
saturation for racemate and dextro- tartrate (2), in the other 
to that for racemate and laevo-tartrate. Schematically : 

A^) Kacemate 4- (^Tartrate 
(^Tartrate + ^Tartrate (i) / 

27 MS) Racemate -f^Tartrate 

The results of such measurements of solubility may best 
be represented by a model, in which the quantities of the 
two salts are measured along two perpendicular planes, 
whose line of intersection constitutes the axis of tempera- 
ture. The solubility curves then lie between the two 
planes, and any projection of them can be obtained, such as 
Fig. 21. 

The essential difference between the formation of a 
double salt and a racemate now appears in the symmetry 
-possessed by the latter case, due to the exactly equal 
solubilities of the optical antipodes, whereas the components 
of a double salt have in general different solubilities. The 
number of measurements needed to prepare the model is 
for this reason smaller in the case of the racemate, and it 
is only necessary to experiment with the racemate, or 
inactive mixture, and one of the antipodes. This is the 
case with rubidium racemate : only here the phenomenon 
is reversed as compared with double-salt formation, for the 
formation of racemate takes place at lower temperatures, 
and in the equation 

<Rb 2 C 4 6 H 4 ) 2 . 4 H 2 ^ QKb 2 C 4 6 H 4 + 

the left-hand side represents the form stable at low tem- 
peratures (below 40-4). Thus, what in the preceding 
equation came on the left, the tartrate mixture (i), comes 


here on the right, and conversely the mixture of racemate 
with the respective tartrates (2) and (3) comes here on the 

Solubility measurements of the racemate-tartrate mixture 
gave per 100 molecules of water, expressed in molecules 
of Kb 2 C 4 O 6 H 4 : 

Te mperature r- Tar fr ate I" Tartrate 





r + l 





Fig. 21 is the projection of a model that represents these 
data. T is the axis of temperature ; dextro-tartrate is 

Fig. 21. 

measured upwards, laevo-tartrate downwards, so that R r/ B 
represents the data just given, while AB and BC correspond 
to those that are also drawn on Fig. 20, and refer respec- 

1 The way in which the relative excess of one antipode approaches zero 
is very characteristic of the transition temperature. 


tively to racemate without excess of either active tartrate, 
and to the tartrate mixture. To complete the figure in 
the plane of the dextro-tartrate, the line RR' is drawn for 
the solubility of the dextro-tartrate alone, from the data 

25 ioo H 2 10.9 Rb. 2 C,0 6 H 4 

52-5 ij n-79 

We may now, on account of the solubility of the two 
antipodes, draw on the same figure S"B and ss' as ex- 
pressing saturation with racemate and laevo-tartrate, or 
with the latter alone, and joining these various lines by 
surfaces we have 

area RR'CBR" saturation for dextro-tartrate ; 
SS'CBS" laevo-tartrate; 

R"BS" racemate. 

The bounding lines refer to saturation for two salts ; the 
point B, in which they meet, to saturation for all three. 

According to what precedes, inactive bodies containing 
asymmetric carbon may be divided into three classes : 

1. Those, the most, that appear in racemic form, like 
racemic acid, and whose transition point is so far removed 
from ordinary temperatures that they appear from the 
inactive solution practically only as racemoids ; the race- 
moid here has a much smaller solubility than the inactive 

2. Others, rarer, that appear split up, like gulonic lactone ; 
here the racemoid has a much greater solubility than the 
inactive mixture. 

3. The third category includes the rarest cases, so far 
observed only in sodium-ammonium racemate, ammonium- 
bimalate, rubidium, and potassium racemates and methyl- 
mannosid, in which, according to the temperature, cases 
i and 2 meet, so that a transition point separates the 
regions of the two phenomena. 

(/3) Separation by forming salts with active acids and 
bases. A method very suitable for separation, when dealing 
with bases or acids, is to combine with suitable active 


substances to form salts. The method was first applied in 
the case of racemic acid by Pasteur, who saturated the acid 
with cinchonine and found that the cinchonine salt of the 
laevo-acid crystallized first ; this has then obviously a 
smaller, and therefore a different solubility \ in accordance 
with the general rule that, so soon as asymmetrical optical 
antipodes combine with the same asymmetrical active sub- 
stance, a compound results no longer answering to a pair 
of reflected images, as appears from Fig. 22. 

The two larger triangles represent dextro- and laevo- 
tartaric acid, the two smaller, cinchonine ; the dextro and 
laevo combinations cannot be arranged 
symmetrically to one another, and con- 
sequently differ in their properties like 
ordinary isomers, e. g. in their solubility. 
Direct measurements of solubility have 
not yet been carried out for such a case, 
so that the question is still undecided 
whether racemism is possible in it, with 
a transition temperature in the former sense 2 . If so, the 
phenomena would be strictly comparable with the behaviour 
of double salts. 

(b) Methods of Separation depending on Chemical Action. 

Separation by means of enzymes and organisms. Separa- 
tion by means of enzymes and organisms is really the same, 
for the function of an organism in such cases can be traced 
to the action of peculiar substances formed in the organism, 
and given off by it, to which the name of enzymes has been 
given. An especially interesting example of this is to be 
found in the isolation effected by Buchner 3 of the enzyme 
of fermentation, zymase, from yeast, by breaking up the 
yeast-cells (by rubbing with glass) and pressing out. If 
we combine this fact with the discovery of Fischer 4 , that 

1 Marckwald, Berl. Ber. 31. 786. 

2 Ladenburg, 1. c. 31. 524, 937, 1969 ; Kiister, 1. c. 31. 1847. 

3 1. c. 31. 568, 1084, 1090. * 1. c. 23. 2620. 

H 2 


only ordinary glucose, i. e. <i-glucose, is capable of fermenta- 
tion, but not its antipode, ^-glucose, so that the latter may 
be prepared from the inactive ^-glucose by fermentation, 
we have a good picture of what is so far known with 
regard to this subject. The mode of action of the enzyme 
is still completely unknown; it may be mentioned, however, 
that in some cases (conversion of maltose into glucose l ) the 
process is reversible, the possibility resting on fine details 
of constitution 2 . 

2. Single Carbon Linkages and Compounds with more 

than one Asymmetric Carbon Atom. 
The principle of free rotation. Whilst so far we have 
dealt practically only with simple asymmetry, in compounds 
of the type 

CR 1 R 2 R 3 R 4 

we have now to turn to single linkages between several 
carbon atoms and the appearance of multiple asymmetry. 

It is a consequence of the fundamental conception of 
stereochemistry that in such single linkage one carbon 
atom lies at one of the angles of the 
tetrahedron, at whose other angles are 
the three groups combined with the 
other carbon atom ; this is shown in 
Fig. 23 . The line j oining the two carbon 
atoms is thus fixed, leaving the three 
attached groups capable of rotation 
about it. Different positions of these 
groups causes an isomerism that has 
Fig. 23. so far only been observed in rare cases, 

and perhaps not certainly there *\ Thus 
the mutual action of the groups attached to the carbon 
atoms seems in general to determine a single (preferential) 
relative position, corresponding to the only known, actually 

1 Hill, Journ. Chem. Soc. Trans. 1898, 634. 

2 Fischer, Bedeutung der Stereochemie fur die Physiologic ; Zeitschr. f. Phys. 
Chem. 1898, p. 60. 

3 Aberson, Die Apfdsaure der Crassulaceen ; Bert. Ber. 31. 1432. 


prepared compound. This position may be chosen so that 
the six groups combined with the two carbon atoms lie 
opposite one another in pairs, as in Fig. 23. 

Number of isomers in multiple asymmetry. From the 
standpoint thus adopted the number of isomers is only 
increased by single linkage with a second carbon atom 
when a new asymmetric grouping is so introduced, as e. g. 
in a compound of the general type 

C (Rj^RgRgj C (R 4 R 5 R 6 ). 

The number of isomers in this case is two on account of the 
one carbon atom, and is doubled by the presence of the other, 
consequently four. In the same way, if n such carbon atoms 
unite the number becomes 2 n . 

The arrangement and behaviour of these isomers may, 
in the case of two asymmetric carbon atoms, be made clear 
by the use of Kekule's wire model, and may be represented 
on a plane by appropriate projection. Taking for that 
purpose Fig. 23 as starting-point, R 1? R 2 ,R 4 , and R 5 lie on the 
plane of the diagram ; then R 3 may be brought into it by 
rotation upwards, about an axis passing through R : Ro, and 
R 6 by rotation downwards about an axis passing through 
R 4 R 5 . We thus get the substance represented by the 
formula No. i following, and the three isomers are 
Nos. 2, 3, and 4 : 

JVb. i No. 2 No. 3 A T o. 4 

RS R.3 J**3 R.S 

RI OjA< 2 rv^O-bv^ jttj Ort 2 jLv 2 Oi\^ 

R 4 CR 5 R 4 CR 5 R-CR 4 R 5 CR 4 

R 6 R 6 R 6 R 6 

It appears from this mode of representation that the four 
isomers are arranged in pairs 2 and 3, i and 4, each group 
including two configurations behaving as opposite reflected 
images, and are thus the expression of optical antipodes 
like those that occur with a single asymmetric carbon atom. 
This behaviour is still more easily understood by con- 
sidering the optical rotation due to each carbon atom 


separately, which we may write A and B, A and B. 
The rotations then are : 

No. i No. 2 No. 3 No. 4 

A + B -A+B A-B=-(-A+B) -A-B 

= -(A + B). 

These possibilities are easily realized by combining an 
acid and base, each of which occurs in two opposite active 
modifications, such as lactic acid and coniine. The four 
possible salts then are : 

No. i No. 2 No. 3 No. 4 

A+B -A + B A-B -A-B. 

cMactic acid. t?-coniine Z-lactic acid. c?-coniine c?-lactic acid, i-coniine Macticacid. ^-coniine 

As an example of this fourfold isomerism in a single body 
the dibromocinnamic acids may be chosen : 

The two asymmetric carbon atoms are distinguished by 
underlining. It is well known that Liebermann succeeded 
in obtaining from cinnamic acid, C 6 H 5 CHCHCOOH, by 
addition of bromine, an inactive mixture or racemic com- 
pound which could be split up into the pair of the one type. 
The other pair was obtained in a similar way from allo- 
cinnamic acid, which is isomeric with cinnamic acid. 

Inactive undecomposable type. A separate discussion is 
required for the case, which may be regarded as a special 
case of the above, in which the atoms or groups attached to 
the two carbon atoms are similar in pairs, i. e. 

E t = E 4 , E 2 = E 5 , E 3 = E 6 , 
as in tartaric acid, 

in which 

Ej = E 4 = H, E 2 = E 5 = OH, E 3 = E 6 = COOH. 

The four former symbols then become : 

E 3 E,, E 3 E 3 

E t CE 2 E 2 CE 1 E] CE 2 EgCEj 

Ej CE 2 EjCEg EgGEj E 2 CE^ 

E 3 E 3 E 3 E 3 

The rotations corresponding are : 

No. i No. 2 No. 3 No. 4. 

AA o AA=2A + 

since in No. i the groups round the two carbon atoms are 
reflected images of each other, and therefore produce equal 
rotations, but in opposite senses. There arises, therefore, 
an inactive type that is not capable of decomposition, 
Nos. i and 4, which, however, is represented by only 
a single substance, since i and 4 can be made to coincide 
by rotation, and, therefore, symbolize one and the same 
substance. The facts as to tartaric acid completely verify 
the expectation, for we have : 

No. 2 and No. 3. The two active tartaric acids, together 

with racemic acid, their compound. 
No. i or No. 4. Inactive undecomposable tartaric acid. 

Several asymmetric carbon atoms. In this case it is an 
advantage to use, instead of Kekule's metallic model, that 
of Friedlander, in which each carbon tetrahedron is 
represented by four rubber tubes united in the centre, 
whilst little sticks of wood inserted in the tubes render 
possible mutual connexions, or serve to indicate the 
attached atoms or groups by means of coloured balls on 
the ends. The carbon atoms joined together, givin 
a substance of the general formula, 

C(abc), C(de) 9 C(fgh), 

are expressible, according to Fischer's suggestion, by 
ing c upwards about the axis a&, and h downwards about 
the axis fg. Everything is then in one plane, and can be 
shown upon paper. The 2 n = 2 3 = 8 possibilities for the 
cases of triple asymmetry are then as follows : 

IS. I 














































There arise therefore four types, each including a pair of 
antipodes : 

i, 8 2, 7 3, 6 4, 5. 

As an example we may take the pentoses 
and considering only the types, we have 

CH 9 OH 

Nos. (i, 8) 

(2, 7) (3, 6) 












Actually these four types are found, in arabinose, ribose, 
xylose, and lyxose, of which arabinose is known in the two 
oppositely active forms, while of the other three types one 
antipode of each is wanting. 

A definite choice of one of the four formulae, say for 
arabinose, may now be made on the following two 
grounds : 

1. Arabinose is oxydized to a glutaric acid, 


which is active. In this way the possibilities Nos. (i, 8), 
and Nos. (3, 6) are excluded, since they would lead to 
a glutaric acid with symmetrical configuration, and there- 
fore of the inactive undecomposable type. 

2. Glucose and mannose are formed from arabinose by 
replacing the group 


with HCOH 

two isomers of the formula CH 2 OH(CHOH) 4 COH being 
produced ; first, according to Kiliani and Fischer, hydro- 
cyanic acid is added, the iiitril produced converted into 


acid, and the acid into aldehyde. These two isomers, 
glucose and mannose, are converted by oxydation into 
saccharic and mannosaccharic acids, with the formula 
COOH(CHOH) 4 COOH, which are active, a fact only 
compatible with type 2, 7 which leads to the bodies 














whilst type 4, 5 would lead to these : 







f\ Tin 







of which the latter is symmetrical, and belongs to the class 
of inactive undecomposable modifications. 

Whilst therefore the configuration may, reasoning from 
the necessary symmetry of the inactive undecomposable 
type, be traced out into detail, it remains undecided which 
of the two enantiomorphic forms 2 and 7 is to be identified 
with a given arabinose. 

No. 2 No. 7 

CH 2 OH CH 2 OH 





3. Double Linkage and Ring Formation. 

Carbon double linkage may be represented graphically 
by a pair of tetrahedra, as in Fig. 34, which are so connected 
as to use up two bonds on each side. The other four then 


lie in a plane, and the configuration of a compound, 
C(ab)C(cd), is expressible by the formula 

aCb aCb 

II or || 

cCd dCc 

which shows at the same time that there are two possibili- 
ties, and that whether the two atoms or groups ab 
combined with carbon are the same as or 
different from cd. It is well known that 
in this case also isomerism is found, and 
we may discuss this more fully by con- 
sidering fumaric and maleic acid. These 
two acids are expressible by the formulae 


II and || 


and correspond in their behaviour to what 
might be expected from the difference between these con- 

In the first place, we are dealing with an isomerism of 
quite a different kind to that due to a single asymmetric 
carbon atom. The formulae are symmetrical, and so there 
is no optical activity and enantiomorphism of crystalline 
form to be expected ; but, on the other hand, we must not 
expect the exact agreement in properties specific gravity, 
melting point, &c. which is produced by the agreement 
in internal structure when isomerism is due to an 
asymmetric carbon atom. If we follow out the differences 
further, we find them in very good accord with the 
conception of their internal structure arrived at. In the 
first place, one of the acids, maleic, gives an anhydride 
very easily : HC CO 

II >0 

which is in accordance with the proximity of the two 
carboxyl groups in the formula. In the second place, this 


same acid is the stronger, as might be expected from the 
influence exerted by the oxygen of either carboxyl on the 
other carboxyl group, which is close to it. The dissociation 
constants, which are closely related to the strength of an 
acid (Part I, p. 137), are respectively 1 

KMal. = T'I7, Kpum. = '93- 

Thirdly, the acid salt of maleic acid is weaker than that 
of fumaric 2 , just because in the ions of these salts, 


the negative charge opposes further dissociation, and does so 
more strongly in maleic acid, as being nearer. The quantity 
dissociated, found by means of the rate of inversion, is 

v Diss. Maleic Acid Diss. Fumaric Acid 
6 4 0-342 % 0-932 % 

I2 8 0-55 1.52 

256 0-918 2.74 ,, 

Finally the above formulae are confirmed by the results 
of oxydation in aqueous solution. Permanganate gives 
with maleic acid, on taking up hydroxyl, inactive undecom- 
posable tartaric acid, as is to be expected : 


gives TTppnnTT 


whilst fumaric acid gives, under the same circumstances, 
racemic acid : 


II or || 






1 Zeitschr.f. Phys. Chem. 3. 380. 2 1. c. 25. 241. 

I2 4 


Ring formation. Finally we may consider the leading 
conclusion of stereochemistry as applied to the formation 
of rings, and in particular to the case of trimethylene 
dicarboxylic acid, 



Representing such a ring in the following way : 

with the six attached groups at i to 6, there are these 
possibilities : 

No. 2. 


No 3. 

II /|\ COOH COOH /\\ II 
II Xl /H 





Formula i represents a symmetrical, and therefore inactive 
form ; Nos. 2 and 3 a pair of asymmetrical images, 
consequently active. Actually two inactive isomers are 
known, of which perhaps one is the unseparated mixture 
or racemic compound of Nos. 2 and 3. 

4. Stereochemistry of other Elements. 

The spatial considerations applied with so much success 
to the carbon compounds may be extended to compounds of 
other elements. We will here only mention what refers to 
nitrogenous bodies. 

First must be considered the possibility of separation 
into optical antipodes, suggested by Le Bel in the case of 
isobutylpropylethylmethylammonium chloride : unfortu- 
nately the researches on this point are only fragmentary, 


and it is at present uncertain if the observed activity is to 
be ascribed to an active ammonium chloride. 

On the other hand, isomerism in bodies of the type 





is fully established, both in the acetoximes which are 
derived from an asymmetrical ketone, i. e. 

ACB C 6 H 5 CC 6 H 4 C1 

ft such as li- 


and in the aldoximes which correspond to the special case 
B = H, a simple example being acetaldoxime : 



The two isomers thus arising are inactive, but differ 
physically like ordinary isomers, and chemically, amongst 
other ways in that one compound easily gives off water 
with formation of a nitril, which may be supposed due to 
proximity of the hydroxyl to hydrogen. The structural 
difference is therefore fco be expressed by the formulae 


II and || 


which recall those of fumaric and maleic acid. 

Concluding remark on stereochemistry. As in deter- 
mining constitutions in the earlier cases intramolecular 
displacement of the atoms may lead to wrong conclusions, 
making it desirable for a satisfactory determination to 
have two lines of argument, so in the more exact deter- 
mination of configuration in stereochemistry the same 
thing must be borne in mind. Such direct changes have 
indeed often been observed within the region of stereo- 


chemistry, e. g. the active bromosuccinic esters gradually 
lose their power of rotation, by becoming converted into 
a mixture or compound of the optical antipodes 1 . This 
phenomenon is one especially noticed in halogen, and chiefly 
in bromine compounds, and probably explains why reactions 
carried out on such compounds have often given quite 
unexpected results ; one of the most remarkable instances 
is that active bromosuccinic acid gives one malic acid with 
alkalies, but with silver oxide the optical antipode 2 . At 
present, therefore, these halogen compounds should, as far 
as possible, be excluded from determinations of constitution 
in this province. 


The phenomenon of tautomerism is that a compound 
shows an apparently different constitution according to the 
reagent that acts on it. This has been observed, e. g., in 
aceto-acetic ether, C 2 H 3 OCH 2 C0 2 C 2 H 5 , and led to two 
constitutional formulae, one proposed by Frankland, 

H 3 CCOCH 2 C0 2 C 2 H 5 , 
and the other supported by Geuther, 

H 3 CC(OH) = CHC0 2 C 2 H 5 . 

The first formula is based, inter alia, on the fact that 
with potash a decomposition takes place with formation of 


H 3 CCOCH 3 . 

The second is in accord with the formation inter alia of the 
body : 

H 3 CC = CHC0 2 C 2 H 5 

N(C 2 H 5 ) 2 , 

by the action of diethylamine, HN(C 2 H 5 ) 2 . 

The explanation of this peculiar behaviour was sought 

1 Walden, Berl. Ber. 31. 1416. 2 1. c. 


for by Laar 1 in a movement of the hydrogen atom, indicated 
in the following symbol : 

H.CCH C0 2 C 2 H 5 

' O 

It has of late been suggested 2 that tautomerism corre- 
sponds to a mixture of two isomers in chemical equilibrium. 
Fundamentally these two conceptions agree, only the first 
is the molecular-mechanical view of the fact formulated in 
the latter. We will therefore recount the facts on which 
the latter formulation is based. 

In the first place the supposed isomerism has been 
observed in certain cases in closely allied bodies. Wilhelm 
Wislicenus 3 observed this, e. g., in formylphenylacetic ether, 
which is very similar to aceto-acetic ether. Obtained from 
ethyl formate and phenyl-acetate : 

HC0 2 C 2 H 5 + C C H 5 CH 2 C0 2 C 2 H 5 = HCOC (C 6 H 5 ) 

HC0 2 C 2 H 5 + C 2 H 5 OH, 

the body in question behaves as if containing hydroxyl : 
HCOH = C (C 6 H 5 ) C0 2 C 2 H 6 , 

e. g. it gives an addition product with phenyl-cyanate, 
OCNC G H 5 . On melting (indefinitely between 60 and 70) 
an alteration sets in ; solidification does not take place, and 
the substance, now liquid at ordinary temperatures, retains 
indeed the same molecular weight, but colours with ferric 
chloride, and behaves like an aldehyde. This modification 
is known as the ' aldo-form ' (the former as ' enol- '), and is 
represented by the formula 

HCOCH(C 6 H 5 )C0 2 C 2 H 5 . 

The reconversion is accomplished by potassium carbonate, 
in which the solid compound at once dissolves, but the 

1 Berl Bcr. 18. 648, 19. 730. 

2 8. Traube, 1. c. 29. 1715. 

3 1. c. 28. 767 ; Guthzeit, 1. c. 31. 2753. 


liquid only slowly and with transformation into the other ; 
on treatment with acid and shaking out with ether the 
enol-form is recovered. It may be added that the two 
formulae might be confirmed by the fact that the first 
compound should exist in two isomeric states, like fumaric 
and maleic acids : 


II and r || 

C 6 H 5 CCO 2 C 2 H 5 C 6 H 5 CC0 2 C 2 H 5 , 

while the second, containing an asymmetric carbon atom, 
should split into two optical antipodes. 

As a second item towards the explanation of the be- 
haviour of tautomeric compounds, Klister's 1 proof should 
be mentioned that a mixture in chemical equilibrium can 
actually be obtained by means of partial transformation. 
The body considered was hexachlorketopentane, C 5 C1 6 O, 
which is obtainable in two isomers, distinguished as /3- and 
y-compounds, with the probable constitutional formulae : 


Actually the /3-compound is distinguished by forming with 
aniline in alcoholic solution a slightly soluble anilide, 
which is made use of to determine the proportions of the 

On heating, each isomer suffers a transformation leading 
to a mixture in chemical equilibrium that shows all the 
characteristics of a tautomeric compound. The formation 
of this mixture was followed out at 210-5. Measured 
amounts of /3-pentane were heated for a measured time (), 
and after sudden cooling the fraction (x) converted into 

1 Zeitschr.f. Phys. Chem. 18. 161. 






following table contains 


results : 

t (in hours) 


-llog (1-2-591*) 












o 206 















The final quantity 0-386 of y-pentane was practically the 
same as the 0-387 left on heating pure y-pentane for 
sixteen hours at 210-5, and the course of the reaction 
corresponds to the equation that may be expected in such 
a transformation : 

dx ' f . 7 
= k(i-x)-k L x, 

that is, two unimolecular processes going in opposite senses. 
In the final condition 

= o, so that k (i x) = k L x, 


and since 

x = 0-386, Jc 1 k i -591, 

so that 


= /c(i- 


-\ log (1-3-5910) 

must be constant, as the third column of the above table 

It may be added that Walden observed the same state of 
equilibrium between two isomers in the spontaneous forma- 
tion of racemates. Active bromosuccinic ethyl ester, e. g., 
gradually loses its activity by transformation into a mixture 



or compound of the optical isomers. The equation to the 
change is simplified on account of the similarity of the two 
opposing reactions, for 

k = fc l} 

dx . . 

Finally the behaviour of aceto-acetic ether, apparently due 
to mutual transformation at ordinary temperatures, is most 
peculiar. The phenomenon has been accurately studied, 
especially by R. Schiff l . He concluded, on the ground of 
the formation of the two isomers, 

H 3 CCOCHC0 2 C 2 H 5 H 3 CCOH = CC0 2 C 2 H 5 

| and 

C 6 H 5 CHNHC 6 H 5 C 6 H 5 CHNHC 6 H 5 

Melting point 78 Melting point 103 

with the aid of benzaniline, 

C 6 H 3 CH = NC C H 5 , 

that aceto-acetic ether is a mixture of two isomers : 
H 3 CCOCH 2 C0 2 C 2 H 5 and H 3 CCOH = CHCO 2 C 2 H 5 , 

i. e. the keto- and enol-forms respectively. This view was 
confirmed by the fact that different preparations did not 
yield the isomeric benzaniline derivatives in the same 
proportion; one obtained from Kahlbaum giving the pure 
enol-form, another from Marquardt giving a mixture. 

The measurements of Traube 2 point to the same con- 
clusion, indicating a slow change in the density : 

1-02443 ...... 15 minutes after distillation. 

1.02467 ...... 20 hours later. 

This went hand in hand with a change in the behaviour 

1 Berl. Ber. 31. 603. 

2 1. c. 29. 1715. See especially also Schaum, 1. c. 31. 1964. 


towards ferric chloride. The phenomenon was more striking 
in ethyl-alcoholic solution : 

0-85906 after 10 minutes. 

0-85963 i^ hours. 

0-86072 20 

0-86066 ,, 8 days. 

In chloroform this was not observed : 

1.42277 after 10 minutes. 

1-42273 4 hours. 

It is questionable, however, whether in alcoholic solution 
the process 

C 6 H 10 O 3 + C 2 H 6 O = aCH 3 C0 2 C 2 H 5 

did not occur. 

Whilst, therefore, the latest information points to a 
tautomeric compound being a mixture in equilibrium of 
two isomers, it may be remarked l that such a phenomenon 
is only possible in a liquid or solution. In the solid state 
one or other compound must be the stable form, and the 
two can only coexist at the transition temperature. Liquid 
or dissolved, however, the condition of equilibrium is 
subject to gradual displacement with temperature, so that 
a mixture in equilibrium is possible over a considerable 
interval of temperature. In general, therefore, a solution 
of a pair of isomers near the transition point would behave 
as tautomeric. 

Here we may add the remark that the phenomenon of 
tautomerism, like any other depending on atomic move- 
ments, must vanish at the absolute zero. This, however, 
is only in accordance with the known fact that tautomerism 
does not occur in the solid state. Long before the absolute 
zero all tautomeric compounds would become solid. 

1 Knorr, Berl Ber. 30.2389. 

I 2 



WHILST polymerism is explained as difference in molecular 
weight, and isomerism in molecular structures, there remains 
a third difference in properties to account for, which is 
characterized by the fact that it disappears on conversion 
into the amorphous, i. e. liquid or gaseous state, conse- 
quently on fusion, evaporation, or solution. As example 
may be mentioned the rhombic and monosymmetric forms 
of sulphur, with regard to which it has lately been shown * 
that they form the same molecules S 8 on going into 
solution; along with these are to be placed the various 
crystallized modifications of ammonium nitrate, &c., studied 
by Lehmann; and finally the numerous polymorphic 
minerals, such as calcite and aragonite. 

It is natural to refer these differences to variation in 
molecular grouping, since when the grouping ceases and 
the substance passes into the amorphous state, the differ- 
ences vanish. For that reason polymorphism is most 
analogous to the physical change of state on melting or 
freezing; it is well known that a molecular orientation 
is then destroyed or created, whilst in a polymorphic 
transition one orientation appears at the expense of another. 
And so, if we are to distinguish between physical and 
chemical isomerism, it is convenient to regard polymorphism 
as physical isomerism, and polymerism and isomerism as 
chemical, since they in common rest on a change within 

1 Aronstein and Meihuizen (see p. 60). 


the molecule. The word allotropy may then be retained 
for the phenomena of isomerism in the elements, whether 
they are to be characterized as polymerism, like ozone (O 3 ) 
and oxygen (O 2 ), or polymorphism, like rhombic and mono- 
symmetric sulphur. 

The general remark may be added that, as was explained 
earlier (p. 82). the phenomena of polymerism and isomerism 
are rare in the inorganic province, and the differences 
accompanying similarity of composition there met with 
are almost entirely to be referred to polymerism. It is 
remarkable that the tenacity with which organic com- 
pounds retain a molecular structure which is, nevertheless, 
not that of equilibrium, a property mostly wanting in 
inorganic compounds, is replaced by tenacious retention 
of molecular grouping. Two points of view are advan- 
tageous for complete treatment. First, we will bring 
forward the laws of transition in polymorphic substances. 
Secondly, we may discuss the details of molecular grouping. 


These laws may be deduced from the two facts that the 
polymorphic substances are solids of similar composition, 
and that on passing over into vapour, solution, or the 
fused state, the difference between the two modifications 

We have, however, in the following discussion, to bear 
in mind the conditions of equilibrium, that mutual con- 
version of the two modifications, especially in the case of 
polymorphic minerals, such as calcite and aragonite, does 
not occur ; we have, therefore, to deal, in one of the two 
cases, with a state of apparent equilibrium (Part I, p. 236) 
which is not destroyed even by contact with the other 
modification, and is not to be referred to an extremely 
slow transition. It is natural to associate this rigidity 
with the relatively great hardness of the minerals. 


Let us now put together the chief results with regard 
to mutual conversion ; these refer firstly to cases of poly- 
morphism in which a conversion takes place slowly in one 
direction or the other, due to contact with the other modi- 
fication. In these cases we may clearly describe one 
modification as stable, the other as meta-stable. 

A. The Stable Modification must have the Smaller 
Vapour Pressure and the Smaller Solubility, 

The necessity of this rule follows because only in this 
case would a change from the meta-stable to the stable 
phase necessarily occur by means of the vapour, or 011 
contact with a solvent by means of it. If the reverse 
condition of solubility held, the transition would take 
place in the reverse direction on suitable application of 
a solvent; the direct conversion would then allow of a 
possible cyclic process that would be equivalent to a per- 
petual motion. A direct confirmation of the greater solu- 
bility of the meta-stable variety has been given in the 
case of magnesium-chloride octohydrate *, of which the 
saturated solutions possess the following composition : 

MgCl 2 . 8H 2 O a (stable) MgCl 2 . 11-43 H 2 O (-16 8) 

MgCl 2 . 8 Ho Op (meta-stable) MgCl 2 . 11-04 H 2 O (-16-8) 

B. The Stable Modification must have the Higher 

Melting Point. 

In Fig. 25 the saturation pressure P of the two modi- 
fications I and II is represented as a function of the 
temperature, and I possessing the higher pressure is con- 
sequently the meta-stable phase. If now we draw a third 
curve, III, representing the vapour pressure of the liquid 
substance, the points of intersection, A and B, with I and II 
obviously represent the melting points of the two forms, 
and, as may be seen, A, the melting point of the meta- 

1 Van 't Hoff, Meyerh offer, Zeitschr.f. Phijs. Chem. 27. 87. 



stable form, refers to the lower temperature. Thus, 
e. g., benzophenone in the stable form melts at 48, in the 
meta-stable at 26. 

C. Possibility of a Transition Temperature. 

As in all condensed systems, if a transition takes place 
between polymorphic modifications it must be carried out 
completely at one temperature. 
The direction in which it takes 
place may, however, depend 
on temperature, as may be 
seen from the vapour pressure 
curves. If the curves I and 
II for the two modifications 
show a point of intersection 
A (Fig. 26), then below the 
temperature T A corresponding 
to it the modification to 
which II refers will be formed 

Fig. 25. 

from the other: above that temperature the state of 
things is reversed, and consequently on passing that point 

Fig. 26. 

Fig. 27. 

a total conversion takes place, as is the case with 
rhombic sulphur at 95-6, when it is converted into mono- 
symmetric. Lehmann has proposed to call phases such 


as those of sulphur, for which there is a transition tem- 
perature, enantiotropic ; the others, for which the reaction 
will go in one sense only, such as benzophenone, mono- 
tropic. The latter phenomenon may obviously result from 
the transition point lying above the melting points of 
both modifications ; thus Fig. 27 represents enantiotropy, 
Fig. 25 monotropy ; in each figure curves I and II refer 
to the vapour pressure of the two solid phases, III to that 
of the liquid. 

D. When there is a Transition Temperature, the Modi- 
fication stable at Low Temperatures is formed from 
the other with Evolution of Heat. 

The relations of vapour pressure, as shown in Fig. 26, 
give this conclusion at once, with the aid of the second law, 
in the form 

Here V is the increase of volume on evaporation, i. e. prac- 
tically the volume of the vapour; at the transition point 
this is the same for the two modifications, since they have 
the same vapour pressure. Further, as the figure shows : 

dP n > dP l9 

where II is the modification stable at low temperatures ; 

q n > q l or q n - q, = R 
is positive. 

But now, since q n and g x are the latent heats of evapo- 
ration of the stable and meta-stable forms respectively, 
q u q l is the heat evolved on formation of the former 
from the latter ; consequently the form stable at low tem- 
peratures is produced from the other with evolution 
of heat. 

This law is very well illustrated by the researches of 


Bellati and Romanese on ammonium nitrate 1 . This suit- 
stance freezes at 168 in the regular system; becomes 
rhomboheclric at i 24, with an evolution of n-86 calories 
per kilogram ; at 82-5 becomes rhombic, with an evolution 
of 5-33 calories; and at 31 rhombic with a different axial 
ratio, with evolution of 5-02 calories. 

E. Polymorphic Modifications have a Constant Ratio of 
Solubilities, proportional to the Differential Co- 
efficients of the Saturation Pressures, in Solvents 
which take up so little of the Substance, that the 
Laws of Dilute Solutions are applicable. 

This relation follows from the equation (Part I, p. 36) 

dlogO _ _Q_ 

which unites the concentration of the solution or vapour 
with the heat Q absorbed when one kilogram-molecule is 
dissolved or evaporated (without doing external work). 
For the two polymorphic bodies we have 


dT <zT* 

dlOgCu _ QlT 

dT = 2T 2 ' 

(Hog (Cn/CQ _ Qn-Q, 

where R is the heat of transition. 
Integrated, this gives : 

The constant is given by the fact that at the transition 

* Lehmann, Molekularphysik, p. 159. 


temperature P the concentrations (7 [5 C u are equal, so that 


Here, however, the second term is a quantity independent 
of the solvent, depending only on the heat of transition R, 
and which, therefore, remains unaltered even if C l , C u refer 
to vapour, so that their ratio is that of the saturation 

This law, which might easily be proved for, say, calcite 
and aragonite, or for the sulphur modifications, retains 
a certain applicability even to isomers more widely 
separated, which do not become identical on solution. 
Carnelley and Thomas l found for isomeric benzene 
derivatives (para- and meta-nitraniline) the following two 
empirical laws : 

1. Of isomeric compounds, that with the lower melting 
point has the greater solubility. 

2. The ratio of solubilities is independent of the solvent. 
The latter rule corresponds literally with our conclusion 

as applied to solutions. The former equally corresponds 
to the fact of higher vapour pressure shown above to 
accompany lower melting point and greater solubility. 

As is to be expected, however, the rule thus found is not 
universal, since it does not apply to polymorphic bodies, 
and so Walker and Wood 2 found, e. g. for ortho-, meta-, and 
para-oxybenzoic acid in water and benzene, the following 

Ortho Meta 

Ortho Meta Para 

Meta Para 

Water .... 0-264 1-337 0-765 0-197 1-75 

Benzene . . . 0-97 0-0121 0-0052 80-2 2-33 


The answer to the question as to molecular arrangement 
involves two essentially distinct problems, one with regard 
to the relative position of molecule and molecule, the other 

1 Journ. Chem. Soc. Trans. 1888, p. 782. 2 1. c. 1898, p. 618. 



as to the orientation of the molecule in its surroundings. 
Let us take as a definite example the oxide of magnesium, 
periclase (MgO), whose molecules are to be thought of in 
a determinate arrangement, belonging to the regular 
system. Then we may ask whether the line joining an 
atom of magnesium and one of oxygen is parallel to one of 
the axes of the regular system, or perhaps corresponds to 
the direction of an octahedral edge, &c. 

The two problems are treated separately in what follows, 
as : 

A. Relative position of molecule centres in the crystalline 

B. Orientation of the molecules in the crystal. 

A. Relative Position of Molecule Centres in the 
Crystalline Figure. 

i. The Fundamental Law of Geometrical Crystallography 1 
(F. C. Neumann). 

Observation shows that normally grown crystals have 
the form of polyhedra, whose flat bounding faces cut each 
other at definite an- 
gles, while the area 
of the faces depends 
on accidental circum- 
stances. The crystal- 
line form of a body is 
therefore determined 
by the angles, i. e. by 
the relative position 
of the faces by which 
it is bounded. With 
regard to this position, 
however, observation 

shows further that 
not every possibility is 

1 Groth, PJnjsikalische Krystallographie, 3rd ed. Leipzig, 1894. 

Fig. 28. 
found, but that the form of 


a crystalline polyhedron follows the law usually known as 
the ' law of rational indices/ 

This law states that any possible plane in a crystal can 
be determined by means of four planes, no three of which 
are parallel: in what way is best shown in Fig. 28 
by drawing three of the planes about the point o, viz. 
XOY, YOZ, zox, and the fourth as ABC. The position of a fifth 
plane, say A'B'O', is conditioned by the fact that the ratio 
of the so-called indices 

OA' OB' oc' 

OA ' OB oc 

can only be given by whole numbers, such as 
mm: p say 2:3: 5 

or can only be rational. 

In describing a crystalline form therefore, it is sufficient 
to give the angles made by a suitable group of four planes : 

ZOY = a, ZOX = /3, YOX = y, 

as axial angles, and the ratio 

OA : OB : oc = a : b : c 

as axial distances, so that the whole crystal is defined by 
means of five elements. 

2. Attempt at Explanation of the Geometrical Law by 
Arrangement of Molecule Centres (Frankeiiheim, Bravais). 

A very simple conception, probable also on other grounds 
to be explained later, gives a plausible meaning to the 
above geometrical law. This consists in the assumption of 
a regular arrangement of the crystalline centres, such that 
they are alike and parallel throughout the crystal. As 
previously the position of four non-parallel planes sufficed 
for the entire structure, so here the position of four 
molecule centres, j, 2, 3, 4 (Fig. 29), not lying on one plane. 
The axial angles then correspond to the angles between 


Fig. 29. 

a, I, and c, the axial distances to the lengths of a, b, and c. 

The condition that the arrangement round, e. g., point 4 

should be similar and parallel to that 

round i is satisfied by choosing the point 5 

in a direction and distance from 2 parallel 

and equal to b. In the same way 6, 7, 

and 8 are arrived at, and so an elementary 

parallelepiped completed. This may then 

be developed into a regular point system 

(Fig. 30) by extending the line a to an 

equal extent, to obtain a new point, and 

so on. Such an arrangement of molecule 

centres would reproduce the relations of a crystalline 

figure 1 . 

If, now, the bounding faces of the crystal are planes in 
which molecule centres are contained, they may pass either 
through 2, 3, 4 or through i, 2, 3 or through i, 2, 4 or 
through i , 3, 4. With any other 
choice, such as 3, 5, 6, the law 
of rational indices obviously 
holds, since each index cor- 
responds to a row of points 
starting from i with a definite 
number of steps between them. 


Relations of Symmetry 
in Crystals. 

Just as a crystal is not cap- 
able of assuming any form, 
so the relations of symmetry Fig 30p 

of these forms are restricted, 

and from the above fundamental law of geometrical 
crystallography Hessel and Gadolin deduced all the 
actually observed relations. We cannot give the deduction 
here in full 2 , but it may be shown how the law of rational 

1 Groth, Physikalische Krystallograplde, 3rd ed., 1894, p. 248. 

2 Ibid., 1. c. p. 311. 


indices restricts the possibilities of symmetry. In the first 
place, by reference to Fig. 28 we see that there need be no 
symmetry at all, in which case we have the unsym metric 
(tri clinic) system. 

One plane of symmetry may be present, but it is limited 
as to position; if e.g. it passes through zox, then the 
plane symmetrical to ABC would have rational indices 
with respect to ox and oz, but not in general with respect 
to OY ; it is therefore a subsidiary condition that OY 
should be perpendicular to the plane of xoz. These are 
the conditions for the non- symmetric system. 

One axis of symmetry may be present, i. e. a line such 
that rotation about it through an angle less than 360 
brings the crystal into a position identical with its 
original position. If e.g. the axis occupies the position 
oz (Fig. 28), the law of rational indices is satisfied when 
XOY = 90 and OA = OB ; such axes are therefore, for the 
same reason, subjected to certain conditions; they can 
only be double, triple, quadruple, or sextuple, i. e. the form 
may recover its original position on rotation through 1 80, 
120, 90, or 60, i.e. two, three, four, or six times in 
a circle. These are the conditions for the rhombic, 
trigonal, tetragonal, and hexagonal systems. 

A third possibility is that the crystal on rotation comes 
into a position symmetrical to its original position ; it 
is then said to have an axis and plane of compound 

In this way, taking into account the law of rational 
indices, thirty-two classes, exactly corresponding to the 
facts, are arrived at ; and these are grouped in the seven 
crystalline systems : 
I. Triclinic system. 

1. Holohedry. Example: calcium thiosulphate (CaS 2 O P> 
. 6 H 2 0). 

2. Hemihedry : one double axis and one plane of com- 
pound symmetry at right angles to it. Example : eZ-mono- 
strontium tartrate (Sr(C 4 H 5 O e ) 2 . 2H 2 O). 


II. Monoclinic system. 

1. Hemimorphy: one double axis of symmetry. Ex- 
ample: cane sugar (C 12 H 22 O n ). 

2. Hemihedry : one plane of symmetry. Example : 
potassium tetrathionate (K.,S 4 O 6 ). 

3. Holohedry : one plane of symmetry and a double axis 
of symmetry at right angles to it. Example : sulphur (S 8 ). 

III. Rhombic system. 

1. Hemihedry: three double axes of symmetry at right 
angles. Example: magnesium sulphate (MgS0 4 . 7H 2 O). 

2. Hemimorphy : one double axis and two planes parallel 
to it and at right angles to one another. Example : struvite 
(MgNH 4 P0 4 .6H 2 0). 

3. Holohedry : three planes of symmetry at right angles, 
and three double axes of symmetry at right angles. 
Example : sulphur (S 8 ). 

IV. Tetragonal system (one quadruple axis). 

1. Sphenoidic tetartohedry : one plane of compound 
symmetry at right angles to the quadruple axis. Not 
yet actually observed. 

2. Hemimorphic hemihedry. Example: wulfenite 
(PbMo0 4 ). 

3. Sphenoidic hemihedry. As in class i ; in addition, 
two double axes of symmetry at right angles to one another 
in the plane of compound symmetry, and two planes of 
symmetry which cut in the quadruple axis and bisect the 
angles of the double axes. Example : (CuFeS 2 ). 

4. Trapezohedric hemihedry: four double axes in the 
plane at right angles to the quadruple axis. Example : 
nickel sulphate (NiSO 4 . 6H 2 O). 

5. Pyramidal hemihedry: one plane of symmetry at 
right angles to the quadruple axis. Example: scheelite 
(CaW0 4 ). 

6. Hemimorphic holohedry : four planes of symmetry 
cutting one another in the quadruple axis. Example : silver 
fluoride (AgF . H 2 O). 

7. Holohedry: in addition to the above, four double 


axes of symmetry and one plane of symmetry, all at right 
angles to the quadruple axis. Example : tinstone (SnO 2 ). 

V. Trigonal system (triple axis of symmetry). 
i . Hemimorphic tetartohedry. Example : sodium per- 
iodate (NaIO 4 . 3H 2 O). 

2. Rhombohedric tetartohedry: the triple axis is also 
the sextuple axis of compound symmetry. Example : 
dioptase (CuH 2 SiO 4 ). 

3. Trapezohedric tetartohedry: three double axes in a 
plane at right angles to the triple axis. Example : quartz 

4. Bipyramidal tetartohedry: a plane of symmetry at 
right angles to the triple axis. Not yet actually observed. 

5. Hemimorphic hernihedry : three planes of symmetry, 
cutting one another in the triple axis. Example : tourmaline. 

6. Rhombohedric hemihedry : besides the foregoing, 
three double axes of symmetry in the plane at right angles 
to the triple axis. Example : corundum (A1 2 O 3 ). 

7. Bipyramidal hemihedry : like No. 5, but in addition 
a plane of symmetry and six double axes of symmetry, 
all at right angles to the triple axis. Not yet actually 

VI. Hexagonal system (sextuple axis of symmetry). 

1. Hemimorphic hemihedry. Example: nepheline 
(Na s Al 8 Si 9 34 ). 

2. Trapezohedric hemihedry : six double axes in a plane 
at right angles to the sextuple axis. Example : antimonyl- 
baric tartrate 4- potassium nitrate ((C 4 H 4 O G SbO) 2 Ba . KN0 3 ). 

3. Pyramidal hemihedry : one plane of symmetry at 
right angles to the sextuple axis. Example : chlorapatite 
(Ca 5 Cl(P0 4 ) 3 ). 

4. Hexagonal hemimorphy : six planes of symmetry 
cutting one another in the sextuple axis. Example : silver 
iodide (Agl). 

5. Holohedry : like No. 4, but in addition one plane of sym- 
metry and six double axes of symmetry, all at right angles 
to the sextuple axis. Example : beryl ( AL,Be 3 (SiO 3 ) ). 


VII. Cubical system. 

1. Tetartohedry : three equal rectangular double axes 
of symmetry, and four equal triple axes equally inclined to 
the double axes. Example : sodium chlorate (NaC10 3 ). 

2. Plagihedric hemihedry: three equal rectangular 
quadruple axes of symmetry, four triple axes as in the 
preceding, and six double axes bisecting the angles between 
the quadruple axes. Example : cuprite (Cu 2 0). 

3. Pentagonal hemihedry: as in No. i, but in addition 
three planes of symmetry at right angles to the double 
axes. Example : pyrites (FeSJ. 

4. Tetrahedric hemihedry: planes of symmetry as in 
No. 3, but in addition six planes which bisect the angles 
between the former planes. 

5. Holohedry : three equal rectangular quadruple axes 
of symmetry, four triple and six double axes as before, in 
addition all the planes of symmetry of classes 3 and 4. 
Example : galena (PbS). 

4. Sohnckes Point Systems. 

The species of regular gratings in space arrived at by 
Bravais and Frankenheim, on the assumption of an equal 
and parallel arrangement throughout the crystal, satisfy 
the law of rational indices, but do not allow of obtaining 
all the thirty-two classes of forms described above; they 
lead indeed to seven systems, but only to fourteen classes l . 

In the year 1879 Sohncke 2 extended this conception by 
rejecting the restricting condition involved in it, that the 
arrangement round each molecule in the crystal is not only 
similar, but parallel. The latter is not necessary with 
regard to the actual observations on crystals ; or rather it 
is definitely not always the case. The observation of 
Baumhauer on calcite, for example, is conclusive on this 
point. Place a prismatic cleavage block (Fig. 31) firmly 

1 Sohncke, Pogg. Ann. 1867, 132. 75. 

2 Entwickelung einer Theorie der Krystalstruktur, Leipzig. 



with an obtuse edge horizontal (best by fitting it into a 
corresponding cut in a board), so that the long diagonal ce 
of the right end-face c d ef is also horizontal, put the blade 
of a knife at a at right angles to the upper edge, and press 
it gradually into the crystal; then 
the part of the crystal to the right of 
the knife will be deformed in such 
a way that when the knife reaches the 
middle edge it will have assumed ex- 
actly the form of an oppositely-placed 
- 3i half-rhombohedron. The part of the 

calcite occupying the new position is 

quite homogeneous physically, but has its plane of cleavage 
parallel to c e g instead of c e d as formerly. 

Taking into account this possibility of an arrangement 
which is similar round each molecule, but not parallel, 
Sohncke arrived at 65 regular point systems (extended 
by Schonflies and Fedorow to 230) which in general consist 
of several congruent interlaced spatial 'gratings/ They 
fall, according to their symmetry, into seven systems, like 
the spatial gratings, and yield exactly the 32 classes of 
possible crystallographic forms given by the fundamental 
law of geometrical crystallography 1 . 

5. The Fundamental Law of Physical Crystallography. 

If Sohncke's theory corresponds, on the one hand, to the 
geometrical relationships of crystals, it allows, on the other, 
an attempt at explanation of their physical behaviour. 
Symmetry of external form is referred to that of the 
internal structure, and explains at once why symmetry 
of physical properties coincides with that of the external 
form, which, it is well known, is the content of the funda- 
mental law of physical crystallography. 

Take, as example, the hardness of rock salt. This shows 

1 Groth, Physikalische Krystallographie, pp. 268, 277. 


three equal minima in the direction of the axes (edges of 
the cube) and four equal maxima at right angles to the 
octahedral faces ; the figure which represents the hardness 
in all directions round a point by radii proportional to it 
has the symmetry of the cube form, i.e. three planes of 
symmetry at right angles to one another, and six others 
which pass through the lines of intersection and bisect the 
angles between the first three planes. 

6. Molecular Compounds. 

A concluding remark may be appended here. The pos- 
sibility of measuring molecular weights, due to the theory 
of solutions, leads to the conclusion that the molecules are 
usually of the smallest size that is in agreement with the 
chemical data. In crystals, too, so far as the theory of 
solid solutions (p. 70) has been applied, no higher molecular 
complexes seem to occur. By the side of this the fact 
appears somewhat surprising that in hydrates, e. g. Na 2 SO 4 
. ioH 2 O, molecules are found possessing a high degree of 
complexity. These seem, however, restricted to the solid 
state, and certainly highly complex liquid molecules have 
not been found to exist ; on the contrary, everything points 
to such hydrates breaking down in solution. The inter- 
laced spatial gratings of Sohncke's point systems allow 
a conception of such hydrates, according to which the 
different molecules would belong to different gratings. It 
is important for this view that some such hydrates can 
lose their water of crystallization without destruction of 
the crystalline form, although their internal physical 
properties are changed ; they take up water again on being 
placed in their former conditions. Mallard 1 and Klein * 
observed this phenomenon with certain aqueous silicates, 
the zeoliths. Lately the same thing has been observed by 

1 Bull, de la Soc. mineral, de France, 5. 255. 

2 Ztitschr.f. Krystallographie, 9. 38. 

K 2 


Tammann 1 with platinum double cyanides, whose com- 
position can vary between MgPtC 4 N 4 6-25 and 6'8H 2 O 
without loss of crystalline form. 

B. Orientation of the Molecules in the Crystal. 

j. Relations between Symmetry in the Crystal 
and in the Molecule. 

Since the application of the methods for determining 
molecular weights to solids (p. 70) has made it highly pro- 
bable that the solid state is not characterized by the con- 
dition of complex molecules, the way is open to bring the 
configuration of molecules, as already developed (p. 90), 
into relation with crystalline form, and especially with the 
arrangement of molecules in regular point systems (p. 141) 
which give a plausible molecular-mechanical explanation 
of that form. 

We may first mention what, of such researches, refers 
to the relations of symmetry in the molecule and in the 
crystalline structure. The leading result is the necessary 
conclusion pointed out by Pasteur, that a whole made up 
of similar asymmetric molecules cannot possess symmetry. 
What in his time could not be sharply formulated has since 
the development of stereochemistry become immediately 
clear 2 . Consider as unsymmetrical grouping such as one 
assumes on presence of an asymmetric carbon atom, e. g. in 
ammonium bimalate : 


in which the four different atomic complexes or atoms, 
COOHCH 2 , H, OH, COONH 4 , are thought of as in tetra- 
hedral arrangement round the carbon atom distinguished 
by underlining (Fig. 16, p. 105); a regular arrangement of 

1 Buxlioevden and Tammann, Zeitschr. f. anorg. Chem. 1897, p. 319; 
Wied. Ann. 1897, p. 16. See also Riniie, Neues Jahrb. Mineral. 1899, I. p. i ; 
Joannis, Comptes Rend. 109. 965, no. 238. 

3 Becke, Min. u.petrogr. Mitteilungen von Tschermak, 10. 414, 12. 256. 


such figures can never lead to a symmetrical whole. So 
the crystalline form too of this salt is a rhombic form 
made unsymmetric by the presence of the semi-pyramids, 
in front, above on the left, below on the right ; behind, 
above on the right, below on the left. Though the external 
form in such cases does not always show the absence of 
internal symmetry, other indications of the asymmetry are 
present as erosion figures 1 . 

If the molecule is symmetrical, still the arrangement in 
the crystal may lead to an unsymmetrical whole, as is the 
case, e.g., with quartz (SiO 2 ), 
whose molecular constitution is /\""f\~""?\ 

, i , C / W \/ \ 

symmetrical, as appears from 
its triatomicity, as well as from 
its inactivity in the fused state. 
Just for this case, Sohncke 2 
worked out a crystalline struc- 
ture that gives an account of the 
enantiomorphic and therefore asymmetric reflected image 
forms of quartz, and the opposite and equal optical activity 
associated with them. Assume first three spatial gratings, 
in which the particles are arranged 
according to the trigonal prism 

(Fig. 32), interlaced as in Fig. 33, Q Q 

in which they are projected on j g* 

the base of the prism ; point i 

requires a rotation of 120 about Q 2< m 3 ^ 
an axis passing normally through ^ 

the centre of the triangle 1,2,3, an( l O. 

a displacement along the latter, ^ 

to come into the position of point 2 ; Fi q 

the latter, a similar screw move- 
ment, to become 3 ; finally, a repetition of the movement gives 
a fourth point, whose projection, however, coincides with 
that of i. If then we draw a circle through the projections 

1 See the discussion Trube, Walden, Berl. Ber. 29. 1692, 2447 ; 30. 98, 288. 

2 Groth, Physik. Krystallographiej 1894, p. 262. 


i, 2, 3, and on it, as base, erect a cylinder at right angles to 
the plane of projection, it is plain that the particles wilt lie 
on its surface in spiral order, which is right-handed (like 
a corkscrew) when 3 is twice as far above the plane of i 
as 2 is, but left-handed when 3 is twice as far below that 
plane as 2 is. The spirals show these opposite senses 
whether looked at from above or below ; such a right- and 
left-handed spiral can in no way be made to coincide, for 
one is the reflected image of the other, and they are 
therefore called enantiomorphic. 

Finally, symmetry, if it exists in the molecule, may be 
raised by the crystal 1 . Two tetragonal pyramids, placed 
symmetrically on the same base, if they have the form of 
half octahedra, become an octahedron and so possess the 
symmetry of the regular system with nine planes of 
symmetry, while the individual molecules have only four. 

The symmetry present in the molecule may therefore be 
either raised or lowered by arrangement in the crystal ; for 
a greater number of compounds in which the opposing 
influences, tending to raise and lower the symmetry, partly 
neutralize, the remark of Buys-Ballot holds that the simpler 
the composition, and consequently greater the symmetry 
of the molecule, the greater the symmetry of the crystal. 
The following table 2 allows of forming an opinion on this 
point : 


Elements Diatonic 

Triatonic Tetratonic 

63 20 

So / 68i / 



/o 5 % 
, 35 
i 5 

> 5 11 

Hexagonal . 

35 11 J 9T 11 

Tetragonal ... 

Rhombic .... 

5 11 3 

Monoclinic ... 


Triclinic .... 

Mean Symmetry .... 8-4 7.9 6-2 4.7 

As an expression of the entire condition, we have chosen 
the mean symmetry, obtained by multiplying each percen- 

1 See, among others, Barlow, Zeilschr.f. Krysl. 1898. 

2 Zeitschr.f. Phys. Chum. 14. i, 548. 


tage by the number of planes of symmetry corresponding, 
viz % 9, 7, 5, 3, i, o, and dividing the sum of these products 
by TOO. Like all statistical relations this has the disad- 
vantage that the application to particular cases often fails. 

2. Influence of Changes in the Molecule on the 
Crystalline Form. 

In the preceding, attention was paid to the direct relation 
between symmetry or the want of it in the molecule and 
the crystal. A second line of research is that of the 
influence which changes in the molecule exert on corre- 
sponding changes of form. Let us take first the smallest 
changes which show themselves in the orientation or 
relative position in the molecule, as polymorphism, while 
the molecule as a whole remains the same. Here we have 
clearly a point of departure in judging of the relations 
between molecular configuration and crystalline structure, 
since the same configuration is associated with two different 
figures. The observations collected on this subject are, 
however, so far of a purely empirical character, and amount 
to no more than the important remark that, when the 
crystalline form of a polymorphic substance changes, there 
is often a striking equality in the angular and axial 
relations to be noted J . Let us consider examples. 

Potassium nitrate, potassium sulphate, and calcium 
carbonate, which can all take hexagonal form, i. e. with 
the angle 60, show in their possible rhombic forms prism 
angles not very different from 60 actually 60 36', 59 36', 
and 63 44' in the three cases. 

Basic copper nitrate, Cu.^(OH) 3 N0 3 , which is found in 
rhombic form with the axial ratio 

a : b : c = 0-9217 : i : 1-1562, 
is converted into a monoclinic form with almost the same 


a : b : c = 0-919 : i : 1-1402, 

1 Arzruni's collection in Graham-Otto, 1893, 74. 


and an axial angle not very far from 90 : 
/3 = 9433'- 

Finally the silicate, H 2 Mg 19 Si 8 34 F 4 , is found as humite 
(rhombic) : 

a : b : c = 1-0803 : i : 4-4013, 

as chondrodite (monoclinic) : 

a : b : c = 1-0803 : i : 3*1438 /3 = 90, 

and as clinohumite (monoclinic) : 

a : b : c = 1-0803 : i : 5-6588 /3 = 90. 

This equality of form on change of structure would 
perhaps come out more strikingly if the comparison of 
angles and lengths were made, not at the ordinary tempera- 
ture, but at the transition point, since then the two forms 
have just the same stability, a fact which may be due to 
mechanical causes. In any case it is to be remarked that 
this relation between the two forms is associated with the 
possibility of a transition without upsetting the internal 
connexions of the substance. Thus, e. g., boracite passes at 
266 into the regular form without any loss of transparency, 
but only with a sudden change of optical properties, 
becoming singly refracting. 

Isorneric change constitutes a deeper modification in the 
molecule. Accordingly, the accompanying change mostly 
complete in the symmetry of the molecule corresponds to 
a complete change in molecular structure, without any 
relations that have so far been traced. In one case, 
however, the circumstances are different, viz. in that of 
stereo-isomers with an asymmetric carbon atom. The two 
isomers here present, as we have seen, a difference of con- 
figuration shown in Figs. 16 and 17 (p. 105), and known as 
enantiomorphic. A completely analogous enantiomorphic 
difference is found in the crystalline form, as is shown in 
Fig. 1 8 and 19 (p. 106), for the oppositely active ammonium 


On passage from the active compounds to the racemate, 
in which the enantiomorphic molecules unite, the enantio- 
morphism of crystalline structure is also destroyed, as 
a rule. In the cases hitherto studied the following axial 
ratios have been found : 

Potassium bitartrate, KC 4 H-O 6 , 
Rhombic a : b : c = 0-7168 : i : 0*7373 
Potassium biracemate, (KC 4 H 5 O 6 ) 2 , 
Triclinic a : b : c = 0-7053 : i : 0*7252 
a = 88 36' /3=:io2 22' y=87i6', 
so that at any rate two axial angles are not far from 90. 

If we have here a striking relation between the configura- 
tion of the molecule and crystalline structure, it becomes 
much more so on considering the phenomena of isomorphism, 
which Mitscherlich first put in the right light. That two 
bodies like potassium-magnesium sulphate and rubidium- 
cadmium sulphate, despite their entirely different com- 
position : 

19.4 / Potassium 29-2 / Rubidium 

6 ,, Magnesium 19*3 ,, Cadmium 

I 5*9 > Sulphur n Sulphur 

55-7 ,, Oxygen 38-5 Oxygen 
3 Hydrogen 2 Hydrogen 

should possess the same monoclinic crystalline form, which, 
moreover, indicates so close a similarity in internal structure 
that they form completely homogeneous mixed crystals in 
which one compound has replaced the other, is a most con- 
clusive proof that the molecular-mechanical conception of the 
compounds, attributing to them the analogous formulae, 

K 2 Mg (S0 4 ) 2 6 H 2 0, Rb 2 Cd (SO 4 ) 2 6 H 2 O, 

comes very near to the facts if only schematically. 

It is on this ground of isomorphism that considerations 
may be developed allowing of a deeper insight into the 
structural relations in crystals. 

In the first place we must consider the investigation 
started by Groth on so-called morphotropy, i. e. small 
changes of form which are produced by change of composi- 


tion despite prevailing isomorphism. The following measure- 
ments of Topsoe furnish an example ; all the compounds 
are hexagonal : 


t = (CH 3 ) 3 H 

R 4 =(C 3 H 7 ) 3 H 
R 4 = (C 2 H 5 ) 2 

: I-I075 
: i -ioo2 
: 1-0855 
: 1-0512 
: 1-029 
: 1-025 

4 = (C 2 H 5 ) 3 H ......... i 1-017 

R, = (C 2 H 5 )H 3 ......... 1:0-9955 

Deeper changes are observed in the following cases T : 

PtCl^.2NR i Cl System Axial ratio /3 

R 4 = (CH 3 ) 4 ...... regular 

R 4 --=(CH 3 ) 3 C 2 H 5 .... ,, 

R 4 = (CH 3 ) 2 (C 2 H 5 ) 2 . . . tetragonal 1:1-0875 

R t = (CH 3 )(C 2 H 5 ) 3 ... i : 1.0108 

R 4 = (C 2 H 5 ) 4 ..... monoclinic 0-9875:1:0-9348 90 46' 

Here the influence affecting the form of the crystal has 
changed its system, yet without much alteration in the 
axial ratio or angle. Here plainly we have to do with di- 
or tri-morphism, and the morphotropic action has modified 
the transition temperatures, so that, e. g., with the tetra- 
methyl derivative at the ordinary temperature the regular 
form, but with the dimethyl-diethyl derivative the tetragonal 
form, is the more stable. 

The new researches of Tutton 2 and Muthmann 3 carry us 
still further in the direction of an insight into the position 
of the molecules in a crystal. From the latter work we 
may take some results relating to the tetragonal phosphates 
and arsenates, whose isomorphism was observed by Mit- 
scherlich : 

Substance Molecular volumes a : c a c 

KH 2 P0 4 ...... 58-246 1:0-6640 i 0-664 

KH 2 AsO 4 ...... 62-822 1:0-6633 1-026 0-683 

NH 4 H 2 PO t ..... 64-170 1:0-7124 1-009 0-719 

68-842 i : 07096 i 031 0-732 

1 Arzruni, 1. c. 233. 

2 Journ. Chem. Soc. Trans. 1893. 337, 1894. 628, 1896. 344. 

3 Zeitschr.f. KrystaUographie, 1894. 


We see from these data, first, that in isomorphic compounds 
an equal difference in composition gives an equal difference in 
volume, for the molecular volume (molecular weight divided 
by the density) gives : 

for As P in the potassium salt . . . 4-576 

ammonium ... 4-673 

for NH 4 K in the phosphate .... 5-924 
arsenate 6-020. 

Next, however, and this is the chief result, the axial ratio 
remains almost unaltered on replacement of phosphorus by 
arsenic, but changes by a considerable amount on replace- 
ment of potassium by ammonium. Let us calculate, in 
order to express this behaviour more precisely, the dimen- 
sions of the fundamental parallelepiped, keeping for 
potassium phosphate the values 

a = 6 = i, c = 0-664. 
The volume is then ^ a 2 r, and so 

a-c (= 0-664) : a i 2c i 5^*^46 : 62-822, 

where a t , c l refer to the fundamental parallelepiped of 
potassium arsenate. But since 

Oj : c t = i : 0-6633, 
we get 

&j = 1-026, <?! = 0-683. 

The corresponding values for the other salts are given 
in the fourth and fifth columns of the table. It follows 
that on replacement of potassium by ammonium, there is 
produced essentially only an increase in the height c of the 
fundamental parallelepiped amounting to 0-055 and 0-049 
respectively; while a is practically unaltered, or only by 
the amounts 0-009 an( i 0-005 respectively. On the other 
hand, replacement of phosphorus by arsenic not only 


increases c by 0-019 or * OI 3> but also a by 0*026 or oo22. 
It is then natural to assume that with the configuration 




the line passing through K O P=O gives the direction 
of the principal axis c, so that the enlargement due to 
substitution of NH 4 for K only lengthens the principal 
axis, whereas that due to the substitution of As for P 
produces an expansion in all directions. 


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